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TECHNICAL DRAWING SERIES 


ELEMENTS OF MECHANICAL DRAWING 


THE USE OF INSTRUMENTS; THEORY OF PROJECTION AND ITS APPLICATION 
TO PRACTICE; AND NUMEROUS PROBLEMS INVOLVING BOTH 
THEORY AND PRACTICE 


BY: 


GARDNER C. ANTHONY, A.M., Sc.D. 


PROFESSOR OF DRAWING IN TUFTS COLLEGE AND DEAN OF THE DEPARTMENT OF ENGINEERING 3 
AUTHOR OF ‘‘MACHINE DRAWING’’ AND ‘‘ THE ESSENTIALS OF GEARING’’} MEMBER 
OF THE AMERICAN SOCIETY FOR THE PROMOTION OF ENGINEERING EDUCATION 3 
MEMBER OF AMERICAN SOCIETY OF MECHANICAL ENGINEERS 


REVISED AND ENLARGED EDITION 


BOSTON, U.S.A. 
D. C. HEATH & CO., PUBLISHERS 
1907 





F OPEV \' 
PTV ) _ Copyricut, 1894 anp 1904, 


By GARDNER C. ANTHONY. 


= my 1o I. 28 


Be [LO Man. Kyte Pr, be 


Apis 


Mec a. 


PRHFACE 


THE extended use of the “ Technical Drawing Series” has necessitated a very complete 
revision of the ** Elements of Mechanical Drawing,” resulting in some radical changes in 
the book. In adapting it to so wide a range as would include its use in the Evening 
Drawing School and the Technical College, it has been found necessary to separate the 
instruction from the problems so that it may be useful to instructors who desire a book 
of reference for use in connection with problems and notes of their own. 

The folding plates have been abandoned, and the illustrations are now printed with 
the text, great care having been used to make the references to the cuts either on the 
Same or opposite page. Many illustrations have been added, and all have been redrawn. 

The problems and their lay-out are printed at the end of the book, with numerous 
references to the text. The number and variety have been increased, and include many 
of a practical character suitable for elementary courses, all of which have received the 
test of the classroom. The student should be required to master the principles before 
attempting to solve the problems, receiving such instruction as his special case may 
demand. By this means individual instruction may be given to large classes. 

The method recommended for finishing drawings by leaving the construction lines in 
pencil, neither inking nor erasing them, has been found efficient for the following reasons: 
—It enables the instructor to follow the methods and reasoning of the student; it teaches 
neatness and care in the execution by preventing the free use of an eraser on the com- 
pleted drawing; it is a great saving of time. 

Ml 


157893 


IV PREFACE 


The Third Angle Method of projection is used exclusively, in accordance with the 
best modern practice. Some college instructors have objected to this because it is not 
commonly adopted in treatises on Descriptive Geometry; but the author has used it 
instead of the First Angle Method for the teaching of this subject during the past four 
years, and it has caused no difficulty or confusion on the part of the students. 

After a student has acquired the necessary skill in penmanship, the greatest stress 
should be laid on the subject of Projection. He should be taught to regard Graphics as 
a language study, the grammatical construction of which is developed in the Theory of 
Projection. No copying should be permitted, save in learning to use the instruments, and 
the subject should be taught as an art of expression rather than one of pictorial repre- 
sentation. Although most people recognize drawing as a medium for conveying thought, 
few appreciate the importance of teaching it as a language. But such it is in the fullest 
sense, possessing a well-defined grammatical construction, rich in varied forms of expres- 
sion, forcible yet simple, and truly universal. 

The author desires to express his thanks for the many suggestions and kindly criti- 
cism made by those who have found this book useful in the classroom, and wie have 
helped to make it what it is. 

GARDNER C. ANTHONY. 


Turts COLLEGE, 
July, 1904. 


TABLE OF CONTENTS 


CHAPTER I 
INSTRUMENTS AND THEIR USE 
ART, PAGE ART, PAGE 
1. The Outfit 1 | 15. Use of Curves . ; ° ° ° . 0 
2. Instruments and Materials 1 | 16. Compasses : : . : . ot he 
3. Drawing Board 1 | 17. The Use of Compasses ° 12 
4. T Square 2 | 18. Dividers : : 14 
5. Use of T Square 2 | 19. Bow-pencil : 16 
6. Triangles . 3 | 20. Bow-pen . : F : om Jui suelo 
7. Use of T riangles 3 | 21. Bow-spacers. : 7 ‘ : : ul 
8. To Test the Angles of Trian gles 5 | 22. The Ruling-pen : ; ; ° yw 
9. Pencils : : 6 | 23. To Sharpen the Ruling-pen é . : ugh Es 
10. To Sharpen the Pencil 6 | 24. Erasers and Erasing . ’ ; : : baer) 
11. Pencilling . : Cet 20.0 meee. : ; ‘ : : ‘ - 20 
12. Scales . 3 : 8 | 26. Paper ; . ; : . aD PAY 
13. Use of Scales. A Nya. 8 | 27. Miscellaneous Material. : ° ° 20 
14. Curves 3 : : s 10 


CHAPTER II 
GENERAL INSTRUCTION 
28. Preparation of the Paper . - : : . 21 | 33. General Instruction for Inking : . . 28 


29. Character of Lines . ; Be ony ; hed at ote A Lacy st ‘ ; ‘ ‘ ; : oe eta 
30. Shade Lines. : : - ; ; . 23 | 35. Lettering . 3 ; ; é : : . 30 
31. Line Shading . : ; 5 : : Eee Ouh oOuL ible. : : ; ‘ ‘ ‘ : pei 
32. Sections . : q 4 : : : Peers oy Peed tatoberia yea : : , : : , ey 


ART, 


38. 
39. 
40. 


42. 


48. 


TABLE OF CONTENTS 


CHAPTER III 
GEOMETRICAL PROBLEMS 


Introduction to Geometrical Problems 

To bisect a right line or the arc of a circle 

To divide a line into any number of equal 
parts : 


. To draw a perpendicular to a line 


Angles 
To bisect an angle 


. To construct an angle equal to ag given angle : 
. To construct an angle of 45° 


To construct angles of 60°, 30° and 15° 

To construct an equilater al triangle, having 
given a side : : 

To construct an isosceles tr iangle 

To construct a scalene triangle 

Tangent, Secant and Normal 


. To draw a tangent to a circle 
. To lay off an are equal to a given tangent 
. On a tangent to lay off a ee equal to a 


given arc 

To draw a circle through a given ‘point and 
tangent to given lines 

To draw any number of circles tangent to each 
other and to two given lines. 

To draw a tangent to two given circles 

To draw a circle of a given radius aueout to 
two given circles 


. Through three given points not in the same 


straight line to draw a circle . ‘ 
To cireumscribe a circle about a given tr iangle 


. To inscribe a circle within a given triangle 
. To inscribe an equilateral triangle within a 


circle 


PAGE 
39 
36 


36 
36 
38 
38 
38 
38 
38 
39 
39 
39 
40 


40 
41 


41 
42 


42 
42 


45 
43 
43 
4 


+4 


ART. 
62. 
63. 
64. 
65. 
66. 


Gir 


73. 


74. 


76. 


77. 


To inscribe a square within a circle : : 

To inscribe a pentagon within a circle 

To inscribe a hexagon within a circle 

To circumscribe a hexagon about.a circle 

To draw a hexagon oes csi a long 
diameter ‘ 

To draw a hexagon having | given a short 
diameter 


. To draw a hexagon having given aside . 


To inscribe an octagon within a given circle 

To circumscribe an “octagon about a circle 

On a given side to construct a regular polygon 
having any number of sides . 

Within an equilateral triangle to draw ‘three 
equal circles tangent to each other and one 
side of the triangle ‘ 

Within an equilateral tr iangle to draw three 
equal circles tangent to each other and two 
sides of the triangle 

Within an equilateral triangle to draw six 
equal circles tangent to each other and the 
sides of the triangle 


. Within a given circle to draw three equal 


circles tangent to each other and the given 
circle 

Within a given circle to draw any number of 
equal circles tangent to each other and the 
given circle : 

About a given circle to circumscribe any. num- 
ber of equal circles tangent to each other 
and the given circle : ; . 


48 
48 
48 
48 
49 


49 


TABLE OF CONTENTS 
CHAPTER IV 
CONIC SECTIONS 
ART, PAGE ART. 
78. The Cone and Cutting Planes. ° . feral 87. Parabola. Second method . 
79. The Ellipse. : : : : ai) 88. Given a parabola to find its axis, focus, and 
80. Ellipse. First method . ‘ : : a directrix ‘ : ; ; . E 
$1. Ellipse. Second method : ‘ : . o4 89. The Hyperbola : 
82. Ellipse. Third method . : : : . 54 90. Hyperbola. First method 
$3. Ellipse. Fourth method ‘ : : Bai, 91. Hyperbola. Second method . 
84. Ellipse. Fifth method . : : ; OO 92. The Equilateral Hyperbola ? 
85. The Parabola . : : ; 5 Peo 93. Method common to all the conic curves . 
86. Parabola. First method. : é : 07 
CHAPTER V 
ORTHOGRAPHIC PROJECTION 
94. Introduction to Orthographic Projection . 61 | 100. The projection of a circle oblique to the coor- 
95. Projection : Filia dinate planes : . é 
96. To determine the projections of an object . 64 | 101. The projection and true length of lines . 
97. Objects oblique to the planes of projection . 67 | 102. Projection of rectangular surface by aatary 
98. Auxiliary planes of projection : yee LO plane , 
99. Views omitted by use of auxiliary plane . 71 | 103. Rules governing the relation of lines and sur- 
faces to the H and V coérdinate planes 
CHAPTER VI 
ISOMETRIC AND OBLIQUE PROJECTION 
104. Introduction to Isometric and Oblique Pro- 112. To make the isometric drawing of a circle 
jection . ; : ‘ : ; . 78 |. 113. The measurement of gree lying in isometric 
105. Axonometric Pr ojection F - : - Att) planes . 
106. Oblique Projection . : : : , - 78 | 114. To make an isometric dr awing of an oblique 
107. Isometric Projection ‘ : : : th timber framed into a horizontal timber 
108. The Isometric Axes ‘ : ‘ ‘ . 79 | 115. Suggestions for special cases . : 
109. The Isometric Scale : . 80) 116. A useful case of axonometric projection 
110. To make the isometric drawing ofacube . 80 | 117. Oblique, or Cabinet Projection 
111. Non-isometric Lines . ‘ , ‘ tel) 


Vil 


PAGE 


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Coo oO OHO 


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Oo He Oo 


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=| 


81 


82 
85 
84 
86 


Vill 


ART. 

118. 
119. 
120. 


123. 
124. 


127. 
128. 
129. 
130. 
131. 
132. 
133. 


140. 
141. 
142. 


143. 
144, 


TABLE OF CONTENTS 


CHAPTER VII 
THE DEVELOPMENT OF SURFACES 


PAGE ART. PASE 
To develop a surface ; . 88 | 121. The development of a cylinder ° . » 92 
To develop a pyramid when cut by a plane . 90 | 122. To develop a cone . A . ° . . 94 
The development of surfaces of revolution . 92 
CHAPTER VIII 
THE INTERSECTION OF SURFACES 

The intersection of cylinders . : : - 96 | 125. The intersection of an sihas to onde a vertical 
The use of auxiliary planes. ‘ 4 - 98 cylinder : ; ; ‘ - 100 
126. The intersection of prisms : ° ° - 101 

CHAPTER IX 
SPIRALS, HELICES, SCREW-THREADS AND BOLT-HEADS 

The Spiral ; ; ; ; ; . 102 | 134. The Double Thread ’ ° . 108 
The Equable Spir al : . 102 | 135. United States Standard V Threads . . 109 
The Equiangular or Logarithmic Spiral. . 103 | 136. Square Threads : ; ; ‘ me 8 | 
Involutes . ; ; ; ; : . 104 | 137. Sphere and Cutting Planes. Lae 

The Helix : , ; . ‘ : . 104 | 138. United States Standard Hexagonal Bolt-head 
Screw-Threads ; ; ° : : . 106 and Nut : eee? 
Conventional V Threads . : : : 108 | 139. Chamfered Head and Nut. mee Pepe os he 

CHAPTER X 
PROBLEMS 

General instructions for a a are the 145. Object oblique to the coérdinate er ° . 182 
problems. 117 | 146. Special problems in projection . 135 
The use of instruments. "Examples for 147. Isometric projection problems ett 
practice 5 : : : . . 117 | 148. Development problems raat 9 
Geometrical pr oblems 4 ; : . 122 | 149. Intersection problems . 144 
Conic section problems. : : : . 126 | 150. Spirals, screw-threads and bolt-head pr oblems 147 
Orthographic projection problems . : . 128 | 151. Miscellaneous . f ; 149 


HLEMEHENTS OF MECHANICAL DRAWING 


CHAPTER I 
INSTRUMENTS AND THEIR USE 


1. The Outfit. It is of the first importance that the student be provided with good 
instruments and drawing material. The cost need not be great if they are selected by one 
experienced in their use. ‘The following list comprises the smallest equipment consistent with 
good work, and a description of these supplies may be found in the succeeding articles. 


2. Instruments and Materials. Drawing Board, 16’’ x 20", page 1. . T Square, 24!’, page 2. 
45° Triangle, 6'’, page 3. 30° 60° Triangle, 10’’, page 8. Scale, 12’, page 8. 4H Siberian 
Pencil, page 6. 6H Siberian Lead, page 6. Pencil Sharpener, page 6. Curves, page 10. A 
set of Drawing Instruments consisting of the following pieces: 54'’ Compasses, page 12; 
5'’ Dividers, page 14; 3/’ Bow-pencil, page 16; 3/’ Bow-pen, page 16; 3’! Bow-spacers, 
page 16, and a 5!’ Ruling Pen, page 17. Pencil Erasing Rubber, page 19. Ink Erasing 
Rubber, page 19. Inks, page 20. Paper, page 20. Pens, Penholder, Penwiper, and 
Tacks, page 20. 


) 3. Drawing Board. A pine board 7’ thick and measuring about 16/’ x 20!’ will suffice 

for the solution of all the problems in this book. As the upper face should be maintained a 

true plane, it is desirable to have two cleats on the back of the board. One of the short 
1 


2 INSTRUMENTS AND THEIR USE 


edges is chosen as the working edge, and made perfectly true. It should be tested from time 
to time, as any unevenness in this will impair the accuracy of a drawing. ‘Thé working edge 
should be placed at the left side, and is the only one which need be used in this elementary 
work. It is customary for draftsmen to use the edge next thg body when drawing long lines 
parallel to the working edge; but should this be done it itate the making of this 
edge true, and having the angle between the two ed y J0® ‘The upper and right- 
hand edges must never be used. 










4. T Square. This consists of a blade securely fastened to a head by means of a clamp 
or screws. As it is necessary that the upper edge of the blade and the inner edge of the head 
be maintained true, glue should never be used in the joint. These edges, together with the work- 

q ing edge of the board, should be examined frequently, as the 
4 — accuracy of the drawing depends primarily on them. The 

working edge of the blade and head should be at right angles. 


9 5. Use of T Square. ‘The head should be held firmly 
against the left-hand edge of the board with the hand in the 
position shown in Fig. 1. By sliding the T square along 
this edge, parallel lines may be drawn. Previous to draw- 
ing a line at the extreme right, slide the hand along the 
blade with sufficient pressure to maintain it in position 
when drawing the line. Never move the T square by the 
blade or draw a line against the lower edge of the blade. 
Do not use the head in contact with the upper or right-hand 





TRIANGLES 4 


edges of the board. ‘The lower edge of the board 
may be used if it is known to be at 90° with the 
working edge. 






6. Triangles. Th 
used by draftsmen are 60°. The 
former has two equal angles % a right angle. 
The latter has angles of 30°, 60° and 90°. Cellu- 
loid is the best material for their construction, 
although wood and rubber are used also. These 
triangles, in combination with the T square, may 
be used to solve a variety of problems, and facility 
in rapid drawing is dependent on the skill ac- 
quired in using them. 


ngles ordinarily 


7. Use of Triangles. The position in which 
triangles are used for drawing lines perpendicu- 
lar to the T square is shown in Fig. 2. As the 
drawing board should be placed so as to permit 
the light to come from the upper left-hand 
corner, this position of the T square and tri- 
angle will avoid the shadow of the blade or of 
the triangle being cast on the line to be drawn. 
Some frequent constructions involving the use of triangles are as follows :— 





Fig. 3. 


4 INSTRUMENTS AND THEIR USE 


E PARALLELS. Lines perpendicular to the T square blade may 
#3 be drawn by sliding the triangle in contact with the blade, as in 
ga] \U™sveorreounes” Kio, 2, When drawn at other angles, or without the aid of a T 
square, slide one triangle on the long edge of another, as indicated 
in Fig. 8, using care to hold the second triangle firmly. The 
first triangle is then slid in contact with the second by means sf 
the first and second fingers. 

PERPENDICULARS. If the lines be perpendicular to the 

T square blade, use the triangle as in Fig. 2; but when the given 
line does not coincide with a position of the T square, use one of 
the positions indicated by Figs. 4 and 5. In Fig. 4, the 60° 
triangle is placed parallel to, but not touching, the given line. 
It is then slid on the 45° triangle to a second position parallel 
to the first and about 3!’ from it. The 46° triangle is then 
moved to the second position and the required line drawn. In 
Fig. 5, the 45° triangle is set parallel to the given line with a short 
side in contact with the 60° triangle. By turning a triangle on its 
right angle, as in Figs. 5 and 6, a perpendicular may be drawn. 

ANGLES. The method for drawing lines at angles of 45°, 30°, 
and 60° with the T square blade is apparent from the figures. 

Angles of 15° and 75° may be obtained by using the 45° and 60° 
triangles with the T square, as in Figs. 6 and 7. Do not draw 
lines within 4’ of the corners of the triangle, and never construct 
angles by drawing lines along adjacent sides of the triangle. 





TEST OF TRIANGLES 


8. To Test the Angles of Triangles. The right angle may be 
tested as follows: Place the triangle on the T square with the 
vertical edge at the right, as in Fig. 8; draw a fine line, AB, in 
contact with this edge, then reverse the triangle so that both edge 
and line may be free from shadow, and move the edge toward the 
line. If they coincide, the angle is 90°. If they do not coincide, 
and the vertex of the angle formed by line and edge is at the top, 
as shown by A, the angle is greater than 90° by half the angle 
BAC. If the vertex of the angle is below, the angle is less than 
90° by half the amount indicated. 

Test or 45° ANGLE. If the 90° angle is known to be correct, 
place the 45° triangle on the T square, as indicated by the dotted 
position, Fig. 9, and draw a line to coincide with the long edge. 
Reverse the triangle so as to bring the second acute angle into the 
position of the first, and if the edge coincides with the line drawn, 
the acute angles are equal and therefore 45°. If the line and edge 
intersect at the bottom, the angle of the triangle at this point of 
intersection is less than 45° by half the amount indicated by AED. 
If they intersect at the top, the angle of the triangle at this point 
is less than 45° by half the amount indicated. 

TEST OF 30° AND 60° ANGLES, If the 90° angle is known to 
be correct, draw a line to coincide with the T square blade, and 
from any point on this line construct an angle of 60°, as in Art. 46, 





es 


HHHH 


\ 


Fig. 10. 





(| femme OR SRA, CIRTENTTTST:Ti1 9D | SS 


Fig. If. 


INSTRUMENTS AND THEIR USE 


page 88. Test the angle by sliding the short edge on the T square 
until the hypothenuse coincides with or intersects the line measuring the 
angle. If the 90° and 60° angles are found to be correct, the third angle 
must be 30°, since the sum of the angles of any triangle is equal to 180°. 


9. Pencils. The grade of lead commonly used by draftsmen is } 
6H; but this is extremely hard, and unless used with great care will 
indent the paper so that the line cannot be erased. A 4H lead requires 
more care and frequent sharpening, but the student will thus acquire 
a lightness of touch which is of value. The double-end holder with 
movable leads has some advantages over the lead pencil; its length 
remains constant; a shorter lead may be used, and leads of different 
gerade may be used in either end. 


10. To Sharpen the Pencil. Remove the wood from both ends by 
means of a sharp knife, exposing about 3/’ of lead. Fig. 10. One 
end should then be sharpened to a conical point, and the other to a 
chisel or wedge-shaped end. ‘This last operation should be done with 
a file or pencil sharpener, but never with a knife. An excellent sharp- 
ener may be made by mounting strips of No. 0 sandpaper, 4’/ long and 
8!’ wide, on a thin piece of wood so that it may be held in the fingers 
without soiling them. Care must be used to prevent the fine lead dust 
on the sharpener from falling on the paper or board, as it will work into 
the surface and be difficult to remove. The chisel end should be 


PENCILLING 7 


wedge-shaped, as indicated in Fig. 10, and the length of the edge should be reduced to about 
one-half the diameter of the lead. ‘This may be done by first making the end slightly conical. 
After finishing the edge on the file or sharpener it is well to rub it on a piece of paper, rolling 
the pencil slightly to remove the sharp corners. Fig. 11 represents the double-end holder with 
leads properly sharpened. If this type is used, a 4H lead may be used for the chisel end, and 
a 6H for the conical end. 


11. Pencilling. Good pencilling is a prerequisite to good inking. <A drawing poorly 
pencilled is seldom well inked. Of first importance is the sharpening of the pencil, and as it 
wears away rapidly it must be sharpened frequently. The chisel end is used for ruling right 
lines, and the conical point for free-hand sketching, lettering and for marking dimensions. 

The pencil should be held vertical, or nearly so, the arm free from the body, and the flat 
edge of the chisel end lightly touching the straight-edge. Do not attempt to draw with the 
pencil point in the angle made by the paper and the edge of the blade or straight-edge. Draw 
from left to right, or from bottom to top of board. In general, lines are drawn from the body, 
the draftsman facing the board when drawing horizontal * lines, and having his right side to 
- the board when drawing long vertical lines. For the drawing of other lines the position of the 
draftsman should be such as to enable him to draw at ease, having a free-arm movement, even 
though it be necessary to draw from the opposite side of the board. The lines should be very 
fine although perfectly clear. 

Learn to economize in the drawing of lines by omitting such portions as may be unneces- 
sary. Invisible lines, which should be dotted when inking, may frequently be pencilled in full. 


* It is customary to speak of lines drawn parallel to the T square as horizontal, and thos? drawn perpendicular to this 
edge as vertical. 


8 INSTRUMENTS AND THEIR USE 


12. Scales. The best type for general drawing 


Ni iii is that shown by Fig. 12. It should be made of box- 
wood, preferably with a white celluloid edge, and — 
being graduated as in the illustration may be used 
for scales of Full, Half, Quarter, and Eighth size. 

It should be used for dimensioning only, and not 
Fig. (2. . ry é 

asa ruler or straight-edge. The measurements are to 

be taken directly from the scale by laying it on the 


drawing,:and not by transferring the distances from 
the scale to the drawing by means of the dividers. 


























Fig. 13. 13. Use of Scales. In laying off dimensions from 
the scale, place it to coincide with the line to be 
measured, and indicate the distances by means of the conical point of the pencil. Place the 
point exactly opposite the required division of the scale, and having lightly marked the paper, 
examine the division on the scale to ascertain if the distance be correct. A steel point is 
also used for laying off dimensions, but in this case care must be observed in puncturing the 
paper, as the hole should be scarcely visible. In making successive measurements on a line, do 
it by addition instead of moving the scale. If it is required to make divisions of 8/’, place the 0 
of the scale to coincide with the first point, and without moving it lay off 3”, W ' 14", ete. 
Since it is not convenient to make all representations of full size, it is Re, to employ 
reduced sizes, or scales. Those most frequently used, together with the method of indicating 
them on the drawing, are as follows : — 


SCALES 9 


SCALE STATEMENT ON THE DRAWING 
Full Size. Scale: Full Size. 
One-half Size. Seale: Half Size, Or Ga LEE: 
One-fourth Size. Scale: Quarter Size, or 3’ = 1 Ft. 
One-eighth Size. Seale <.14 = 1 Ft. 
One-sixteenth Size. Scales se 1 Ft. 
One-twelfth Size. Scale: 1” =1 Ft. 
One-twenty-fourth Size. Sealecoas= = 1p. 
One-forty-eighth Size. Benoit res lakhs 


The first five of these scales may be read from a scale graduated as in Fig. 12, but the last 
three require graduations of forty-eighths of an inch. <A form of scale commonly used, and 
having all of the above graduations, is represented by Fig 13. 

In using the graduations of Fig. 12 for other than full or half size, the following method 
should be employed: In laying off quarter size dimensions, regard each quarter inch division 
on the scale as one inch, and subdivide the quarter inches for the fractional divisions of an 
inch. Thus, if it is required to measure 173" at this scale, lay off 17 quarter inches, and to this 
add three-quarters of the next quarter of an inch. ‘This will be found much more simple than 
the mental operation of obtaining the quarter part of 173/’, and with a little practice this scale 
may be read as rapidly as that illustrated by Fig. 18, and without the confusion arising from 
the combination of a variety of divisions on one scale.. In a similar manner measurements 
may be made at one-eighth and one-sixteenth scale. 

Scales for enlarged sizes are also employed, but they are not in general use. 


10 INSTRUMENTS AND THEIR USE 


14. Curves. for inking lines which are neither right 
lines nor circular arcs, it is necessary to use irregular curves. 
These are made in a variety of forms, but the type having 
the curves of long radii, similar to those illustrated in 
Figs. 14, 15, 16, are the most serviceable. They are com- 
monly constructed of rubber or celluloid. The former is 
poorly adapted to this use, as it is difficult to make pencil 
marks on its surface. Celluloid presents the same difficulty 
if its surface is not roughened. As the manufacturers do 
not furnish the curves with this surface, the draftsman may 
obtain the desired effect by sandpapering them. White 
holly is a suitable material also, and one from which the 
curves may easily be made. 


15. Use of Curves. Many curved lines can be inked by 

means of the compasses, but when the radius is too great, a 

curve should be employed. ‘The most important point to be 

observed in using the curve, is to avoid inking the curved 

line to the full extent of the apparent matching of the curve. 

In Fig. 17, the dotted position of the curve would admit of 

Fig. 14. Fig. 15. Fig. (6. inking the line from C to E; but it will be observed that 
the curve matches the line from B to F. In continuing the 

inking to the right, the curve should be moved into the position shown by the full lines, in 


CURVES 


which it coincides with DE, a portion of the line 
already inked, and would enable the line to be con- 
tinued to the point K. 

If the curved line to be inked is symmetrical with 
respect to an axis, as in the case of the ellipse, Fig. 18, 
proceed as follows: First ink the part of the curve at 
the extremity of the axis, WAV and XBY, by means of 
the compasses, using great care in selecting the radius. 
Next obtain, if possible, a curve to coincide with are 
CW and for a short distance on either side of points C 
and W,so as to insure a perfect copy of the curved 
line. The line should now be drawn from the point W 
to the point C, but never should it pass to the right 
of the point C with the curve in this position. Now 
marking upon the curve a point which coincides with 
the point C, and reproducing this upon the opposite side, 
reverse the curve in order to draw lines to the right 
of C. Similarly ink the lower half of the ellipse. If 
a curve cannot be obtained to coincide with the entire 
are CW, draw as much of the line as possible from the 
point C, and having drawn the four corresponding parts 
of the ellipse, select a second curve to join the curved 
lines with the circular are. 


11 





12 INSTRUMENTS AND THEIR USE 


A spiral is best inked by means of the bow-pen or the compasses. Fig. 19 indicates the 
operations that would be required to ink the curve ACEF. Beginning at the point A, by 
E trial determine the maximum length of an are that can be 
drawn with one setting of the instrument, and ink that part 
of the curve. Suppose this to be AB with center at 1. Simi- 
larly obtain a second radius, but with the center on the con- 
tinuation of the line Bl, as at 2. This will insure a continuous 
curve in that AB and BC will have a common tangent at B. 
In like manner obtain centers 3, 4, 5, inking the curve as the 
centers are found. The lines B2, C3, etc. are not to be 
pencilled, but the direction determined by the eye. 


16. Compasses. Fig. 20 illustrates a compass set of ap- 
proved type. ‘There are three removable parts; the pencil- 
point, pen-point, and lengthening-bar. ‘There is a joint in each leg, and the removable leg 
is provided with a clamp screw. The shank of the removable parts should fit the socket 
closely, and require but a slight effort to remove. It should not drop from the socket, even 
though not clamped by the screw. All joints should work smoothly. The pen is similar in 
design to the ruling-pen. Art. 22, page 17. The lengthening-bar is used to extend the pen 
or pencil-leg for the drawing of large circles. 





17. The Use of Compasses. When first adjusting the compasses for use, place the pen- 
point in the instrument, pushing it firmly against the shoulder, and securely clamping it. 
Adjust the needle-point so that its point coincides with the point of the pen. Once adjusted, - 


COMPASSES 13 


the needle-point should not be changed. Frequently the needle-point is 
made with a conical point at one end and a fine shouldered point at the 
other. The former should never be used, as it makes too large a hole in 
the paper. The alignment of the instrument may be tested by bringing the 
pen and pencil-point together with the legs straight, and also when bent at 
the joints. The points should coincide in both instances. 

To prepare the compasses for pencilling, place a 6H lead in the pencil- 
point, sharpening it to a chisel end as directed for the pencil, Art. 10, page 
6, save that the length of the edge should not be more than ,/’.. Next 
remove the pen-point from the compasses and replace it with the pencil- 
point. Adjust the length of the lead to coincide with the needle-point, 
using great care to adjust the lead so that the direction of its edge shall be 
a tangent to the circle drawn, otherwise the width of the line will be greater 
than that of the chisel edge. 

The compasses are held between the thumb, first and second fingers, 
and rotated from left to right, clockwise. The hand must never be allowed 
to rest on the instrument, as the needle-point would be forced into the paper, 
and the hole for the center made objectionably large. In placing the needle- 
point on a special center, the compasses may be steadied by lightly touch- 
ing the point with a finger of the left hand. The point should be pressed 
lightly into the paper an amount sufficient to prevent slipping, but the hole 
should be scarcely visible. If it is necessary to designate it more clearly, 
sketch a pencilled line about it, but never put the point of the pencil into the hole to blacken 





14 INSTRUMENTS AND THEIR USE 


it. The needle-point must be kept vertical, and likewise 
the pen and pencil points. In drawing a circle, stop the 
line as soon as the circumference is completed, otherwise 
the line may be widened. If several arcs are to be drawn 
from one center, it is better to remove the needle-point from 
the center when changing the radius, as the hole may be- 
~ come badly worn if much pressure is exerted against it. 
The directions for cleaning and filling the pen are the same 
as for the ruling-pen, Art. 22, page 17. 

When using the lengthening-bar, steady the needle-point 
with the left hand while moving the marking-point with the 
right. It is necessary to use care in doing this so as not to 
change the radius. 

Figs. 21 and 22 illustrate the method of holding the 
compasses, and the position of the fingers before and after 
the drawing of a circle. 


18. Dividers. ‘This instrument is similar in design to 
the compasses, but the legs are fixed, and without joints. 
It is used to transfer measurements from one part of a 

Fig. 22. drawing to another, but must not be employed for trans- 
ferring from a scale to the paper. It is also used to divide 
a line into any number of equal parts when the divisions cannot be obtained directly from the 








DIVIDERS 


scale. To do this proceed as follows: Suppose that it is desired to divide the line 
AB, Fig. 23, into five equal parts. Open the dividers to a distance equal to Al, 
about one-fifth of the required space, and, holding them at the joint by the thumb, 
first and second fingers, place one point at A, the extremity of the line to 
be divided, the other point being at 1. By rotating the instrument in opposite 
directions, as though describing a series of semicircles, lay off divisions A 1, 
1 2,2 5,3 4,45. Point 5 being beyond the extremity of the line, the divisions 
are too great, and should be diminished 
by an amount equal to one-fifth of B 5. 
Make a second, or third, trial if neces- 
sary, so that the last division shall fall 
on Bb. If the required number of di- 
visions be even, bisect the given line, 
then bisect these divisions, and so con- 
tinue as long as the remaining divisions 
are even. 

Fig. 24 illustrates a pair of hair- 
spring dividers. The thumbscrew is 
used to obtain a more delicate adjust- 
ment. ‘The ends of the legs should be Fig. 23. 
conical rather than triangular, and must 





15 





Fig. 24. 


be kept sharp. The puncture made by the divider points should be extremely small, but 


sufficiently clear to be readily seen. 


16 INSTRUMENTS AND THEIR USE 


19. Bow-pencil. Fig. 25. This is used for the drawing of circles from the smallest size 
to about ?’’ radius, if it be a 3’’ instrument. Adjust the lead as described for the compasses, 
but the chisel edge must be reduced nearly to a point. 
It should be possible to draw circles having a clean, 
sharp line with a radius of 34/’ or less. The adjust- 
ment for radius may be rapidly made, and with the 
minimum of wear on the screw, by pressing the points 
together, the fingers being placed below the adjusting 
Screw. 


20. Bow-pen. Fig. 26. The range of this 
instrument, and its adjustment, is like that of the bow- 
pencil. In cleaning and filling the pen the same care 
must be used as described for the ruling-pen, Art. 22. 
An excellent test for the setting of the needle-point is 
| as follows: Having adjusted it so that it projects 

Fig. 25. Fig. 26. Fig. 27. sightly below the point of the pen, draw circles with 

the minimum radius of the instrument. If there is a 

tendency to lift the point from the paper, it is too short; but if it is difficult to draw the line 
without forcing the point into the paper, it is too long. 





21. Bow-spacers. Fig. 27. These are used, like the dividers, for the spacing of distances, 
but they have the advantage of being fixed in position so that there is no liability of a change 
in the space by the handling of the instrument. In spacing distances, the instrument is rotated 


THE RULING-PEN 


alternately right and left, as in the case of the dividers, and the 
forefinger should rest on the top of the spacers to steady them. 


22. The Ruling-pen. Fig. 28 is an enlarged view of the instru- 
ment which is used for the inking of all lines other than circular ares. 
The nibs must be of equal length, rounded alike, and equally sharp. 
The pen is filled by inserting the ink between the nibs with a common 
writing pen, or the quill that is furnished with many of the bottled 
inks. ‘The nibs should be opened slightly to admit the ink freely. 
This method of filling should not necessitate wiping the outside of 
the nibs, but great care must be used to insure their being perfectly 
clean. Do not overload the pen, 4’’ of ink being sufficient for the 
ordinary pen. Having filled the pen, nearly close the nibs and try the 
width of the line on a piece of paper, opening or closing the nibs by 
means of the screw to vary the width of the line. The pen should be 
held by means of the thumb, first and second fingers, the thumb- 
screw being held from the body so as to be readily adjusted by means 
of thumb and second finger. Its position should be nearly vertical, 
being inclined slightly to the right. See Fig. 1, page 2. The pen 
should touch the straight-edge lightly, the point being about 3),/’ from 
the edge. If both nibs touch the paper, the line should be of uniform 
width throughout. Varying the pressure of the nibs on the blade 
will cause an unevenness in the line as: 








18 INSTRUMENTS AND THEIR USE 


If both nibs do not touch the paper, because of their being of unequal length, or because the pen is 
inclined from or toward the body, the line will appear as follows : 
If the distance between the nibs be unchanged, the width of the line should remain constant; 
but as the ink dries very rapidly, causing a slight deposit between the nibs, the width of the 
line will be reduced unless care is used to clean the nibs frequently, for which purpose a pen- 
wiper should always be at hand. If the ink fails to flow freely, touch the point lightly to the 
penwiper or the drawing board, and it will dislodge the small deposit at the point which 
prevents the flow. If this fails, clean the pen thoroughly, and refill. Never lay the pen aside 
without cleaning. 





23. To sharpen the Ruling-pen. All first-class pens are properly sharpened when new, 
but with the cheaper qualities this is not commonly the case. The most frequent defect is in 
an insufficient rounding of the point. An excellent test for the ruling-pen is as follows: 
Having filled the pen, adjust it to draw a fine line, and without the aid of a straight-edge draw 
a number of lines with the pen inclined more and more from the body until the inner nib is 
raised sufficiently high to prevent the drawing of a line. Repeat this operation with the pen 
inclined toward the body. If the angles at which the last line is drawn in each case are 
equal, the nibs are of equal length. Next draw lines with the pen inclined to the right and 
to the left about 45°. If the pen moves freely, and draws clean, sharp lines, the point is 
sufficiently rounded. If the pen is too sharp, the cutting of the paper can easily be detected 
by the touch. 

To sharpen the pen, the draftsman should be provided with a fine Arkansas oilstone and 
a magnifying glass, and proceed as follows: Close the nibs, and hold the pen on the stone 


ERASERS AND ERASING , 19 


as for the drawing of lines, inclining it to the right and left, so as to bring the nibs to an even 
length, and round the points properly. Next clean the pen, open the nibs, and grind the out- 
side faces so that the rounded portions shall be equally sharp. In doing this, the pen should 
be held at an angle of about 15° with the oilstone, and rolled slightly from side to side. 
Examine the nibs frequently with a magnifying glass in order to note the disappearance of 
the bright, polished points, which indicates the degree of dullness, using care to avoid a feather 
or wire edge, which would indicate the shortening of the nib, and necessitate a repetition of 
the first operation. It is seldom necessary to touch the inner face of the nibs, but should a 
burr be raised on the inside, it may be removed by clamping a piece of hard paper lightly 
between the nibs, and moving the pen to and fro on the paper. Finally, clean and test the 
pen as directed, using it in connection with the straight-edge for the last test. 


24. Erasers and Erasing. Pencilled lines are best erased by means of a velvet rubber. It 
must be used lightly, and as little as possible. The particles of rubber caused by the erasure 
should be removed by means of a cloth or brush, or possibly by the hand, if it be clean and 
dry. Do not clean the drawing until it is inked, and then with care. A sponge rubber is 
frequently used for this purpose. Avoid rubbing an inked line whenever possible, as the 
softest rubber will tend. to destroy its clearness. : 

Inked lines must be removed with a hard rubber, which is known as an ink eraser, but 
never by scratching the surface with a knife. As India inks dry rapidly, and do not penetrate 
the hard surface of drawing papers, the object in erasing is to remove the ink without injury 
to the paper. By using care and time, any amount of ink may be removed from a drawing. 
Make sure that the ink is very dry before attempting to erase it. Use the hard rubber with a 


20 INSTRUMENTS AND THEIR USE 


light pressure, and confine the erasing as closely to the line as possible. A card or erasing 
shield may be used to protect other lines and surfaces, but this is seldom necessary. 


25. Inks. ‘The bottled inks, preferably Higgins’ Waterproof, are recommended in place 
of the Indian or Chinese stick inks, which require considerable time to prepare, and necessitate 
fresh mixing for each exercise. Red ink is used frequently for center lines, and blue ink for 
dimension lines and steel sections. Care is necessary in the use of colored inks, as they flow 
more freely than the India inks. | 


26. Paper. ‘The paper should be of good quality, reasonably smooth, and of sufficiently 
close texture to admit of considerable erasing. Keuffel and Esser’s Normal paper is a good 
grade for mechanical drawings. The size recommended for the problems of this book is 
11” x 15’’, or one quarter of an imperial sheet, which measures 22 x 380/. 


27. Miscellaneous Material. One-ounce tacks of copper or iron should be used for holding 
the paper to the board. ‘They are better than thumb tacks, in that they may be forced into 
the board so that the heads are almost flush with the surface of the paper, enabling the T 
square to slide freely over them. 

A 170 or 803 Gillott pen with penholder will serve for such figuring and lettering as 
will be required. | 

A piece of thin chamois skin makes the best penwiper, and should always be at hand for 
cleaning the pens and instruments. 


CHAPTER II 
GENERAL INSTRUCTION 


28. Preparation of the Paper. Having placed the drawing board with the long edge next 
the body and the working edge to the left, see that the surface is free from dust and the T 
square and triangles carefully wiped. Next place the long edge of the paper parallel with the 
long edge of the board, the paper being within about 3” of the lower and left-hand edges of 
the board. ‘The T square may be used to insure its being square with the board. Use four 
tacks, one at each corner, forcing them into the board so that they may be flush with the sur- 
face of the paper. ‘The sheet should lie perfectly flat on the board, and whenever the atmos- 
pheric conditions are such as to cause the paper to swell and present an uneven surface, remove 
three of the tacks and again fasten the paper as before, squaring the sheet by the most impor- 
tant line that has been drawn. Instruction for the stretching of paper by glueing to the board 
is given in Art. 37, page 32. 

The space within the margin line will measure 10’ x 14/’, and is to be laid off in the fol- 
lowing manner: The paper being tacked in place, obtain the center of the sheet by placing 
the T square blade so that it will coincide with the opposite diagonal corners of the paper, and 

21 


22 GENERAL INSTRUCTION 


drawing short, fine lines intersecting at C, Fig. 29. To the right and left of this point lay off 
7’; above and below it lay off 5’... Next place the head of the T square firmly against the left- 
hand edge of the board and draw the upper and lower margin lines through the points last 
laid off. The right and left margin hnes should be drawn by means of the 60° triangle used 
in connection with the T square, as shown in Fig. 29. 
If the triangle be too short to draw the entire line 
at one stroke, move the T square into such a posi- 
tion as will enable the remainder of the line to be 
drawn. | 


29. Character of Lines. All pencilled lines should 
be as fine as is consistent with clearness, and the full 
line used as much as possible. If the drawing is to 
be inked, full pencilled lines may be used in the 

Fig, 29. place of dotted or broken lines, wherever confusion 
will not be caused. 

The fine line is used for all visible lines of an object which may not be shaded. 

The shade line is used to assist in the reading of a drawing by suggesting the relation of 
the surfaces. It is used for visible lines only. 

The dotted line is used to designate the invisible lines of an object, and is never shaded. 
It requires more care than all other lines. The dots or dashes should be about #,/’ long and 
the space about 31,/... These lengths must not be increased, however long the line may be. 
Much of the beauty of a drawing is dependent upon the evenness of this class of line. 





CHARACTER OF LINES 23 


Center lines are frequently indicated by full red lines a little finer than the fine black 
lines. If drawn in black, the broken line or dot and dash line is employed. In this case the 


dash may be about 3/’, and the dot about 54,/’ long. Ex- 
tend the line beyond the surface on Sieh it is drawn. 
Dimension lines are indicated in red or blue by a 
very fine line similar to the center line; or in black by 
a series of dashes about 2/’ long. But in either case the 


witness or arrow point must be black and pointed, as in - 


the figure. 

Witness lines are used to indicate the extent of the 
surface measured when the dimension line falls outside 
of the surface. ‘They are made of the same character as 
the dimension lines. 

Border lines seldom require to be heavier than the 
shade lines. 


30. Shade Lines. ‘There is but one reason that can 
justify the use of the shade line, and that is, added clear- 
ness to the representation by indicating the relation of 
the surfaces to one another. By some draftsmen the 
shade line is never used, while by others it is always 
law can be established concerning it, there being times 
and others when it is equally wrong to use it. 


FINE LINE 
SHADE LINE 


DOTTED LINE 


CENTER LINE IF RED 


DIMENSION LINE IF BLACK 


DIMENSION LINE IF 
RED Hoon oa oa ae eee BLUE 


Yi Yi 





L/L, 30. 


used. Both are in error, since no 
when it is a mistake not to use it 


GENERAL INSTRUCTION 


The following conventional method is much used in 
practice, and is recommended because of its simplicity. 
Shade the right-hand and lower edges of all surfaces. 
Do not shade the line of intersection between visible sur- 
faces. ‘The shade line must not encroach upon the surface 


bS 
> 


which it bounds, otherwise the accuracy of the drawing 
would be impaired. Cylindrical surfaces may be outlined 
by fine lines only. Shade the lower right-hand quadrant 
| of outside circles, and the upper left-hand quadrant of 
Fig. 3h. Fig. 32. inside circles. One width of shade line is used for all 

right lines. Never shade pencilled lines or dotted lines. 


In general the shade line is intended to represent 
the surface in shadow, and the light is supposed to come 
from the upper left-hand corner at an angle of 45°; but 
if there are lines of the drawing parallel with the ray of 
light, the angle of the ray may be considered as greater or 


less than 45°. Thus, in Fig. 34, it is better to shade one 

of the 45° lines rather than to draw both fine. Fig. 31 

represents a square block with a square hole in it. ‘This 

; is indicated clearly by the top view alone, although it 
Fig. 33. Fig. 34. does not show the depth. Fig. 32 is a similar object, 


but in this case it is surmounted by a square block of smaller size. The shade lines indicate 
this also, but fail, of course, to give the height. Fig. 33 represents two views of a hexagonal 


SHADE LINES 


pyramid properly shaded. The lines radiating from the center of the top 
view, and the two inside slant lines of the front view, are division lines 
between visible surfaces, and therefore not shaded. | 

In the shading of circles and circular ares it is necessary to avoid the 
sudden transition from the shade to the fine line, and this is accomplished in 
the following manner: Having inked the circle with a fine line, remove 
the point of the compasses from the center, using care not to change the 
radius, and place it below and to the right of the first center, a distance 
equal to the desired width of the shade line. If it is an outside circle, 
draw the second arc on the outside, as in Fig. 35; and if it is an inside 
circle, it should be drawn as in Fig. 36. If the width of the shade line is 
such that the two eccentric arcs are not in contact throughout, the inter- 
vening space may be filled by slightly springing the instrument. Since 
the circular arcs are the first to be inked, care should be used to adopt 
a width of line that will be correct for the shaded right line. 

The shading of Fig. 387 differs from the two preceding in having a 
uniform width of shade line between points A and B, for the outside circle, 
and between C and JD, for the inside circle. ‘The transition from shade to 
fine line is made within an are of about 45° beyond the points indicated. 
This method is not so simple as the preceding, and the improved appear- 
ance will rarely justify its use. 





26 GENERAL INSTRUCTION 















































Fig. 41. 





Fig. 88 represents a rectangular piece with a de- 
pression in the center, all of the corners being rounded 
or filleted. Small ares, such as these, and all circles hav- 
ing radii less than 3!’, should be inked by means of the 
bow-pen, which may be used in the manner described. 
It is better, however, to acquire the skill necessary to 
spring the bow-pen, so that the shading may be done 
without removing the needle-point from the center. 
This is accomplished without changing the position of 
the thumb and first finger, which are used to handle the 
instrument. By a slight pressure of the first finger, 
sufficient to deflect the needle-point leg, the radius is 
slightly reduced when shading an inside circle, and in- 
creased when shading an outside circle. By this method 
the work may be much more rapidly executed. 


31. Line Shading. When it is necessary to suggest 
the character of a surface without a second view, line 
shading may be used. Figs. 39, 40, and 41 illustrate 
good methods for the shading of cylindrical surfaces. 
The lines may be equally spaced, although the appear- 
ance is somewhat improved by increasing the space as 


one approaches the center line, and decreasing the space on the lower half, while increasing 


SECTIONS ek 


the width of the line. In shading a concave surface, as in Fig. 41, the operation is reversed. 
Cylindrical parts of small diameter may be shaded on the lower half only, as in Fig. 40. 
Conical, spherical, and other classes of surfaces may be ee by this method, but the 


rendering of them requires considerable skill, Big ey ats 
and it is beyond the scope of this treatise to A 


consider the various methods. an a ALAA 
a 
\ Gia 








4-+ BOLTS 


an 
ZL 


. 
AX . 
tN NI] 
MX 
~ \ 
\ 






32. Sections. It is frequently necessary 
to make the representation of an object as it 
would appear if cut by a plane, and with the 
portion nearer the observer removed, as in 
Fig. 42. This cut section is indicated by a 
series of parallel lines, usually drawn at an 
angle of 45°. The width of the lines is the 
same as the fine line, and the distance between 














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Fig. 42. 


strument known as a section liner is designed 
to insure accuracy in the spacing, but the beginner should learn to judge this by the eye. In 
doing so, care must be used to avoid making too small a space, and it is desirable to try the 
spacing on a separate sheet before sectioning the drawing. A greater degree of uniformity 
may be obtained by glancing back after drawing every six or eight lines. 


28 


CAST IRON 








INSULATING MATERIAL 


GENERAL INSTRUCTION 





















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DOTTED SECTION 


Fig. 43. 


The different surfaces in the plane of the section are 
indicated by changing the direction of the lnes, as illus- 
trated by Fig. 42, which is the cross section of a piston 
rod packing. Seven distinct surfaces are clearly shown 
by changing the direction and angle of the section lines. 

Differences in the material are also indicated by the 
character of the lines; but there is no general agreement 
as to notation. Fig. 45 illustrates six good types of sec- 
tioning, together with the names of the materials which 
the author has chosen to indicate by them. Whenever 
figures or notes occur in a section, the section lines should 
not be drawn across them. 


33. General Instruction for Inking. Never begin the 
inking of a drawing until the pencilling is completed. 
See that the sheet is free from dust. Always ink the 
circles and circular ares first, beginning with the small 
If the drawing is to have shade lines, shade each 
arc as drawn. ‘To omit the shading of circles until they 
have first been inked in fine lnes. will necessitate almost 
double the time otherwise required. It should be ob- 


arcs. 


served that the width of both fine and heavy lines is determined by the shading of the first 


circular are. 


Next, ink all the full and dotted ruled lines. 


Begin on the upper side of the 


TRACING 29 


sheet and ink all the fine horizontal lines, omitting those lines which are to be shaded. Next, 
ink the vertical lines, beginning with those at the left. This method insures sufficient time 
for the drying of the ink. Do not dwell too long at the end of a line, especially if it be a 
heavy one, as the pressure of the ink in the pen will tend to widen the line. If a series of 
lines radiate from a point, allow sufficient time for the drying of each line, otherwise a blot may 
be made. Finally, ink all lines at other angles and those curved lines requiring the use of 
curves. ‘The same order is to be followed in the inking of the shade lines, evenness being 
secured by ruling them at one time. Do not shade dotted lines. Ink center and dimension 
lines, and put on the figures and notes. Draw the section lines and put on the title. 


34. Tracing. When it is desired to reproduce a drawing, transparent cloth or paper is 
placed above the original and the lines of the drawing traced on the new surface, as though one 
were inking a pencilled drawing. ‘Tracing cloth is usually furnished with one surface glazed 
and the other dull. Either side may be used, but as it is difficult to erase from the dull side 
it is better to ink on the glazed surface. Pencilling must be done on the dull surface. Tracing 
cloth is used frequently in place of paper for original drawings, the pencilled paper drawing be- 
ing traced on the cloth in ink. Copies of this tracing may be made by the blue print process. 

As the cloth absorbs moisture quite rapidly, it shrinks and swells under varying atmos- 
pheric conditions. Because of this, large drawings which require considerable time to complete, 
should be inked in sections, as the cloth will require frequent adjustment in order that its 
surface may be smooth and in contact with the paper drawing. Only the best quality of cloth 
should be used, as the cheaper kinds are improperly sized and absorb ink, causing blots. If 
the ink fails to run freely on the glazed side, dust on the surface a little finely powdered 


30 GENERAL INSTRUCTION 


pumice stone or chalk, rubbing it lightly across the surface with a piece of chamois skin or 
cloth. Use care to remove it completely from the surface. 

Inked lines may be removed from the glazed side by means of a sharp knife and a hard 
rubber, or by dusting a little finely powdered pumice stone on the lines to be erased, and 
briskly rubbing them with the end of the finger or a piece of medium rubber. As the pumice 
becomes discolored replace it with fresh powder. In erasing lines from the dull side use the 
hard rubber. Pencilled lines may be removed by the ordinary dite eraser, or by means of a 
cloth moistened with benzine. 


35. Lettering. The subject of lettering is of such importance to the mechanical draftsman 
that he should adopt some clear type for general use, and acquire proficiency in the free-hand 
rendering of it. While at times it may be necessary to make use of instruments and mechanical 
aids for the construction of letters and figures, usually they may be written free-hand. 

The accompanying alphabets are such as may be recommended to students, and will be 
acceptable in the regular practice of drafting. Their study will afford an excellent free-hand 
exercise, as well as skill in the figuring and lettering of drawings. Both types should be 
written without the aid of instruments. The first is known as the vertical Gothic, and the 
second as the slant Gothic. Large and small capitals may be used in the place of capitals and 
lower case, as illustrated. The small capitals and lower case may be made about two-thirds the ~ 
height of the initial letters. This type is written quite easily by means of a hard wood stick, 
preferably orange or boxwood, sharpened to a point like a pencil, the size of the point being 
varied according to the desired width of the line. In doing this great care should be taken 
that the ink be black and slightly thick. The student is referred to the treatise on “ Letter- 
ing,” of the ‘Technical Drawing Series,” for the further consideration of this subject. 


LETTERING | 


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GENERAL INSTRUCTION 


86. Title. In the lay-out of the problems of this book no provision has been made for the 
title save in the case of applied work, such as machine drawings. It has been the custom of 
the author to have the name of the student printed in the right-hand lower corner just outside 
the margin line, and the plate number similarly placed in the upper right-hand corner. The 
capitals should be ;3;/’ high and the small letters 3’. 

In applied work the title should always be placed in the lower right-hand corner of the 
sheet, inside the margin line. It should designate, first, the name of the mechanism ; second, 
the name of the special detail ; third, the scale ; fourth, the date, which is always that of the 
finishing of the drawing. ‘The draftsman’s name or initials should be printed in small type. in 
the extreme right-hand corner within the margin. 


37. Tinting. The surface of a drawing is colored or tinted for the purpose of making clear 
the divisions, as in map drawing; suggesting the character of the materials represented; or to 
indicate the character of the surface, whether plane or curved; and possibly its relation to other 
surfaces by the casting of shadows. 

THE PAPER should be of proper quality, such as Whatman’s cold pressed, and must be 
stretched by wetting the surface and glueing it to the board in the following manner : 
Having laid the paper on a flat surface, fold over about one-half inch of each edge. Thor- 
oughly wet all of the surface save the folded edges, using a soft sponge for this purpose, but 
do not rub the surface. Next apply mucilage, strong paste, or a light glue, to the underside 
of the folded portion and press this to the board with a slight outward pressure so as to bring 
the surface of the paper close to the board. As the glue should “set” before the paper begins 
to dry and shrink, it is necessary to have the paper very wet, but no puddles must be allowed 


eTENTIN CG aos 


to remain on the surface after the edge is glued. The paper must be allowed to dry gradually 
in a horizontal position, as otherwise the water would tend to moisten the lower edge and 
prevent the drying of the glue. If the paper should dry too rapidly, not allowing sufficient 
time for the glue to “set,” the surface may be moistened again. 

THE CoLor employed in making the wash or tinted surface may be a water color or ground 
India ink, but none of the prepared liquid inks are suitable for the purpose. The color should 
be very light, and when the desired shade is to be dark it should be obtained by applying 
several washes, allowing sufficient time for each to dry. The color must be as free from 
sediment as possible, but since some deposit is liable to take place, the brush should be dipped 
in the clear portion only, and not allowed to touch the bottom of the saucer. 

THE BrusH should be of good size, depending somewhat on the surface to be covered, 
and of such quality that when filled with the color or water, it will have a good point. 

Two classes of tinting are employed, the flat tint of uniform shade, and the graded tint for 
the representation of inclined or curved surfaces. 

THE FLAT Tint. Remove all pencilled lines which are not to be a part of the finished 
drawing, and do all the necessary cleaning of the surface, using the greatest care not to 
roughen the paper. Inking should be done after the tinting, but if for any reason it is 
necessary to ink the drawing first, a waterproof ink must be employed. 

In putting on the color, slightly incline the board to permit of the downward flow of the 
liquid, and, beginning at the upper portion of the drawing, pass lightly from left to right, using 
care just to touch the outline with the color, but not to overrun, and making a somewhat 
narrow horizontal band of color. Advance the color by successive bands, the brush just touch- 
ing the lower edge of the pool of water made by the preceding wash. This lower edge should 


4 GENERAL INSTRUCTION 


iS) 


never be allowed to dry, as it would cause a streak to be made in the tinted surface. Having 
reached the lower edge, use less water in the brush so as to enable a better contact to be made 
with the outline. Finally dry the brush by squeezing it or touching it to a piece of blotting 
paper. It may then be used to absorb the small puddle of color at the bottom edge or corner. 

Avoid touching the tinted surface until it is dry, at which time any corrections that are 
necessary may be made by stippling. ‘This consists in using a comparatively dry brush and 
cross-hatching the surface to be corrected. 

If the surface to be covered is large, it is desirable to apply a wash of clear water before 
applying the color. This dampened surface will prevent the, quick drying of the color and 
insure a more even tint. When necessary to remove the tint from a surface, use a sponge with 
plenty of clean water, and by repeated wettings absorb the color, but do not rub the surface 
of the paper. 

THE GRADED TINT may be applied by several methods, the simplest being to divide the 
surface into narrow bands and apply successive washes, each covering an additional band. If 
the tint is sufficiently light, and the bands narrow, the division line between the bands will not 
be very noticeable, but this may be lessened by the softening of the edges with a comparatively 
dry brush and clean water. 

Another method, which requires considerable dexterity, is to put on a narrow band of the 
darkest tint that may be required, and, instead of removing the surplus water from the edge, 
touch the brush to some clean water and with this lighter tint continue the wash over the 
second band. Continue in this manner until the entire surface is covered. 

There are many modifications of this method, all of which require considerable skill and 
are not to be recommended to the student at this stage of his progress. 


CHAP TH Re ite 
GEOMETRICAL PROBLEMS 


38. WHILE the majority of students are familiar with many of the propositions included 
in this chapter, the study of the methods best adapted to the draftsman is of great importance. 
It is not intended that these problems shall serve as copies, or that the examples relating to 
them, and given on page 122, shall be used for all students; but as reference data, and for the 
purpose of illustrating the draftsman’s methods, they are believed to be an essential part of a 
text-book on Technical Drawing. 

In most cases two methods are given, the first being the ordinary geometrical solution 
requiring the use of a straight-edge and compasses; and the second, the more direct method 
employed by draftsmen, involving the use of a T square and triangles as well as compasses and 
dividers. In the geometrical figures the given and required lines are shown in full heavy 
lines, and the construction in full fine lines. 

It is intended that the student shall construct the propositions by the draftsman’s method 
and then employ the method of the geometrician as a test. If the problems are performed 
with great accuracy, the technical skill acquired in the handling of the instruments will be 
correspondingly great. It is not well to ink geometrical problems, as the precision of the 
pencilling will be impaired. 

35 


36 








Fig. 44, 


Cc 
; Fig. 46, 


PROBLEMS 


39. To bisect a right line, AB, or the arc of acircle, ACB. Fig. 44. 
With centers A and B, and any radius greater than one-half of AB, 
describe arcs 1 and 2. Through the points of intersection of these 
arcs draw a line. Its intersection with the given line AB, and the 
are of the circle ACB, will determine the required points. 

DRAFTSMAN’S METHOD. Obtain the division with the dividers 
as explained in Art. 18, page 14. 


40. To divide a line, AB, into any number of equal parts. Fig. 45. 
Let the required number of divisions be five. Draw AC at any 
angle with AB, and lay off five equal spaces of any length. Connect 
the last point, 5, with B, and draw parallels through the other points 
intersecting AB in points 1’, 2’, 3’ and 4’ which determine the 
required divisions of AB. Art. 7, page 4. 


41. To draw a perpendicular to a line AB. 

CAsE 1. Fig. 46. When the given point C is on the line, and 
at or near the middle of the line. 

From C, with any radius, draw ares 1 and 1, and from the point 
of intersection of these arcs with AB, with any radius greater than 
arc 1, draw arcs 2 and 8. The line drawn through the point of inter- 
section of these arcs and the given point, C, will be the required line. 

CAsE 2. Fig. 47. When the point is on the line, and at or near | 
the extremity of the line. ; 


oy 


GEOMETRICAL PROBLEMS 37 


First Method. Let AB be the given line, and A the given point. 
From A, with any radius, describe arc 1. With center C, and same 
radius, describe are 2. Through C, and the intersection of arcs 1 and 
2, draw CE, and with same radius as before, from intersection of arcs 
1 and 2, describe arc 8. A line drawn through A, and the point of 
intersection of arc 3 and line CE, will be the required perpendicular. 

Second Method. From B, with any radius, describe are 4. From 
point D, with same radius, describe arc 5. From the intersection 
of ares 4 and 5, describe arc 6. From the intersection of ares 4 and 6, 
describe arc 7. The line drawn through this last point of intersec- 
tion, F, and the given point B, will be the required perpendicular. 

CASE 3. Fig. 48. When the point is outside of, and opposite, 
or nearly opposite, the middle of the line. — 

From C, with any radius, describe arcs land 1. From the point 
of intersection of these ares with AB, with same radius, describe arcs 
2 and 3. A line drawn through this point of intersection and the 
given point C, will be the required line. 

CASE 4. Fig. 49. When the point is outside of, and at, or 
near, the extremity of the line. 

From C draw any line CD. Find E, the center of CD, by 
dividers, or by Art. 39. On CD as a diameter, describe a semicircle. 
Through the given point ©, and the intersection of semicircle with 
AB, draw CF, which will be the required perpendicular. 

DRAFTSMAN’S METHOD. See Art. 7, page 3. 





38 








Cc 
Fig. 52. 





PROBLEMS 


42. Angles of 15°, 30°, 45°, 60° and 75°, in either quadrant, may 
be constructed by means of the 60° and 45° triangles used in combina- 
tion with the T square, as described in Art. 7, page 4. 


43. To bisect an angle, ABC. Fig. 50. From B, with as large 
a radius as possible, describe arc 1. From,its points of intersection 
with AB and CD, describe arcs 2 and 8. The line drawn through 
their intersection and B will bisect the given angle. 


44. To construct an angle, FDE, equal to a given angle ABC. 
Fig. 51. Draw DE. From B and D, with equal radii, describe ares 
land 2. From E, with radius equal to chord AC, describe are 3. 
Through D, and point of intersection of arcs 2 and 38, draw DF mak- 
ing the required angle. 


45. To construct an angle of 45° with AB at point A. Fig. 52. 
Through the given point, A, describe a semicircle on AB; draw a 
perpendicular through the center C. A line drawn through the point 
A and intersection of the perpendicular with the semicircle will make 
an angle of 45° with AB. 


46. To construct angles of 60°, 30° and 15° with AB. Fig. 53. 
From the given point A as a center, with any radius, describe are 2. 
From B, with the same radius, describe arc 8. A line drawn through 
A and this point of intersection will make an angle of 60 with AB. 


GEOMETRICAL PROBLEMS 


Through the given point A, describe a semicircle on AB. With 
same radius, from C describe are 1. The line drawn through A and 
this point of intersection will make an angle of 30° with AB. 

Having constructed an angle of 30°, as described, bisect the same, 
and FAB will be the required angle of 15°. In this case it would 
not be necessary to draw the 30° line. 


47. To construct an equilateral triangle having given the side AB. 
Fig. 54. Since the sides are equal, the angles will be equal, and there- 
fore, 60°, the sum of the angles of any triangle being equal to 180°. 
With centers A and B and radius AB, describe ares 1 and 2. From 
the point of intersection, C, draw AC and BC. 

DRAFTSMAN’S MetTHop. AB being drawn with the T square, 
through A and B, with 60° triangle, draw AC and BC. 


48. To construct an isosceles triangle. Fig. 55. Having given 
the base DF and the equal sides DE and EF, from centers D and F, 
draw arcs 1 and 2 with radius equal to the given sides. From the 
point of intersection, E, draw DE and EF. 

If the angle be given, construct FDE and EFD equal to the 
given angle, and draw DE and EF. 


49. To construct a scalene triangle. Tig. 56. Having given the 
sides A’B’, A/C’ and B/C’. Draw AB equal to A’/B’. With centers 
A and B, and radii equal to given sides, draw arcs 2 and 1. Draw 
AC and CB. 





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Fig. 55. 0. 
va 
Lu 8" Fig. 56. 


40) 





PROBLEMS 


50. Tangent, Secant and Normal. Fig. 57. If a line AB cuts a 
curve at two points, it is called a secant. Conceive the line as revoly- 
ing about the point A until the second point of intersection with the 
curve shall coincide with the first; the line will then be in the position 
AC, and called a tangent. AD isa line perpendicular to the curve at 
the point of tangency, and called a normal. 


51. To draw a tangent, AB, to a circle. 

Case 1. When the given point A is on the circle. Fig. 58. 
Draw the radius AC, and erect a perpendicular, AB, at A. 

DRAFTSMAN’S MEerHop. Place the triangle to coincide with center 
C and given point A, as though to draw AC. By means of a second 
triangle used as a base, turn the first triangle into the second position 
and draw AB perpendicular to AC. 

CASE 2. When the point is on the circle and the center not 
accessible. Tig. 59. From the given point, A, with any radius, 
describe arcs 1 and 1. Place the edge of the triangle to coincide with 
points B and D. Draw-a parallel line through A. 

CASE 8. When the given point, B, is without the circle. Fig. 60. 
Two tangents may be drawn. On BC as a diameter, describe arc 1; its 
intersection with the circle at A and D will be the points of tangency. 
The angle BAC, inscribed in the semicircle, will be 90°. 


GEOMETRICAL PROBLEMS 


DRAFTSMAN’S METHOD. From the given point, B, draw BA 
touching the circle. Through the center, C, draw a perpendicular to 
AB. A will be the point of tangency. In like manner obtain BD. 


52. To lay off an arc equal to a given tangent. Tig. 61. Let AB 
be the given tangent and AD the are. From the point B step off 
equal spaces with the dividers or bow-spacers until a point of the 
dividers is at or near the point of tangency A. Reverse the motion of 
the dividers, stepping off an equal number of spaces on the curve. 

When several arcs are tangent at the same point and it is desired 
to lay off the length of their common tangent on each, the following 
approximation may be used provided the greatest arc does not exceed 
60°. On AB lay off AC equal to one-quarter of AB, and from C as a 
center describe the are DBEF; the arcs AD, AE and AF will closely 
approximate the given tangent AB. 


53. On a tangent to lay off a length equal to a given arc. Tig. 62. 
If AD be the given arc, draw the chord of this are and continue it to C, 
AC being equal to one-half of AD. From C as a center, describe the 
are DB intersecting the tangent at B. AB will be the required length 
or the rectification of the are. This approximation should not be used 
for arcs greater than 60°. 

The method of spacing the distance by the dividers may be 
employed in this case as in the previous one, Art. 52. 


4] 








PROBLEMS 


54. Todraw a circle through a given point, A, and tangent to given 
lines, AB and BD. Fig. 63. Since the circle is to be tangent to AB 
and BD, its center must lie upon the bisector of the angle DBA; and 
because it is to be tangent to AB at the point A, its center must le 
on the perpendicular to ABat A. Bisect the angle DBA, and through 
the point A draw AC perpendicular to AB. C will be the center of 
the circle, and AC its radius. The draftsman’s method may be used 
for obtaining the perpendicular AC, 


55. To draw any number of circles tangent to each other and to 
two given lines, AB and AD. Fig. 64. Bisect angle DAB, and with 
any radius, HK, draw a circle tangent to ABand AD. From Kk draw 
KE perpendicular to Als, and with radius EK describe arc 4. Through 
KF draw FC perpendicular to AB. C will be the center and FC the 
radius of the second circle. Repeat the process. 

If the radius of the first circle be given, draw a parallel to AD 
distant from it equal to the given radius. The intersection of this 
line, HM, with this bisector of the angle ABD will be the required 
center. 


56. To draw a tangent to two given circles. Fig. 65. Join A and 
C, the centers of the given circles. From D lay off DH equal to AF. 
With center C and radius CH draw arc 1. From A draw a tangent 
to this are. Art. 51, Case 38, page 40. Through B, the point of tan- 


GEOMETRICAL PROBLEMS 


gency, draw CE, and through A draw AF parallel to CE. E and F 
will be the points of tangency, and EF the tangent. 


57. To draw a circle of a given radius, R, tangent to two given 
circles having centers BandC. Fig. 66. From centers Band C draw 
indefinitely, in any direction, lines BF and CE. Lay off HE and KF 
equal to the given radius R, and through F and E, from centers B 
and C, describe arcs 1 and 2 intersecting at A, the required center. 
Since these ares intersect in a second center, there will be two solu- 
tions to this problem. 


58. Through three given points, A, B and D, not in the same 
straight line, to draw a circle. Fig. 67. Bisect the imaginary chords 
AB and BD. The point of intersection, C, of the bisecting lines, 
will be the required center. 


59. To circumscribe a circle about a given triangle, ABC. T[ig. 68. 
Bisect two of the sides, as AC and BC. The point of intersection of 
these lines will be the center of the required circle. Draw a circle 
through A, B and C. This problem is identical with the preceding. 

If the hypothenuse AC should pass though the center D, the 
angle ABC would be a right angle. See also Art. 41, case 4, 
page 37. 


43 





44 





PROBLEMS 


60. To inscribe a circle within a given triangle, ABC. Fig. 69. 
Bisect two of the angles, as CAB and ABC. The point of intersection 
of these lines, D, will be the center of the required circle. From this 
point draw a circle tangent to AC, CB and AB. 


61. To inscribe an equilateral triangle within a circle. Fig. 70. 
From the point C draw are 1 with a radius equal to that of the circle. 
From its intersection with the circle, and with the same radius, draw 
are 2. From center C, and with chord CB as a radius, describe are 8, 
and connect points A, B and C, which will give the required triangle. 

DRAFTSMAN’S METHOD. From point C, on the vertical diameter 
CD, draw CA and CB with the 60° triangle. With the T square draw 
AB to complete the triangle. 


62. To inscribe a square within a circle. Fig. 71. Draw any 
diameter BD. Draw a second diameter, AC, perpendicular to it. 
Connect points A, B, C and D to complete the square. 

DRAFTSMAN’S MetHop. With the 45° triangle draw perpendicu- 
lar diameters AC and BD, and connect points A, B, C and D. 


63. To inscribe a pentagon within a circle. Fig. 72. Draw any 
diameter, GF, and a radius AK perpendicular to it. Bisect KF, and, 
with H as a center, and a radius AH, describe arc 3. With center A 
and radius AL describe arc 4. AB is the side of a pentagon. Obtain 


GEOMETRICAL PROBLEMS 


the remaining points by describing ares 5, 6 and 7 with same radius. 
Connect points A, B, C, D and E to obtain the required pentagon. 
DRAFTSMAN’S METHOD. Estimate an arc equal to one-fifth of 
the circumference, and with the dividers step off this length, dividing 
the circle into five parts and correcting the are as directed for the 
division of a line. See Art. 18, page 14. Connect the points. 


64. To inscribe a hexagon within a circle. Fig. 73. Draw any 
diameter FC. With centers F and C and radius equal to that of the 
circle draw ares 1 and 2. Connect the points of intersection A, B, C, 
D, E and F to obtain the required hexagon. Observe that the angles 
at the center, as BKC, are 60°. 

DRAFTSMAN’S METHOD. Draw a horizontal diameter FC. With 
60° triangle draw diameter EB. Draw AB and ED, and with triangle 
draw BC, FE, AF and CD. 


65. To circumscribe a hexagon about a circle. Fig. 74. Draw 
any diameter, AD. With H as center and radius equal to that of the 
circle, describe arc 1. Bisect the are HLN, and through L draw AB 
parallel to HN. With center K and radius AK describe circle ACE. 
In this inscribe a hexagon by Art. 64. 

DRAFTSMAN’S METHOD. With 60° triangle draw diameters AD 
and EB, and with same triangle draw sides AB and ED, EF and BC, 
AF and CD, each tangent to the given circle. 





46 








PROBLEMS 


66. To draw a hexagon having given a long diameter, AD. Fig. 
75. Bisect AD. With K as center, describe circle ACE, and in this 
inscribe a hexagon by Art. 64. 

DRAFTSMAN’S MetHop. With dividers find center K. Through 
A and D with 60° triangle draw AB and DE. With same triangle 
draw BE, and through D and A draw CD and AF. Draw BC and FE 
to complete the hexagon. ‘To obtain B without finding K draw a line 
through D at an angle of 50° with AD intersecting a 60° line through A. 


67. To draw a hexagon having given a short diameter, GH. 
Vig. 76. Bisect GH. From G draw a perpendicular GF. From K 
draw FKC at 30° with Gk. ‘Through F and with center K draw 
circle FBD. Inscribe a hexagon by Art. 64. 

DRAFTSMAN’S METHOD. With dividers find center K. With 60° 
triangle draw FC, and through G, K and H draw perpendiculars FE, 
AD and BC. Draw the sides FA and DC, AB and DE. 


68. To draw a hexagon having given a side, AB. Fig. TT. With 
centers A and B, and radius AB, describe ares 1 and 2. From their 
intersection, K, with same radius describe circle AEC. Inscribe a 
hexagon by Art. 64. 

DRAFTSMAN 'S METHOD. ‘Through A and B with 60° triangle 
draw AD and BE, and through their intersection, K, draw FC... Draw 
FA, BC; FE, DC and ED: . 


GEOMETRICAL PROBLEMS | 47 


69. To inscribe an octagon within a given circle, ACEG. Fig. 78. 
Draw any diameter GC. At center, and perpendicular to GC, draw 
AK. Bisect AKG and AKC. Connect the points of intersection 
with the circle. 

DrRAFTSMAN’S MEetHop. With 45° triangle and T square draw 
diameters AE, GC, FB and HD, and connect their extremities. 


70. To circumscribe an octagon about a circle, ABCD. Fig. 79. 
Draw the perpendicular diameters AC and BD. With centers A, B, 
C and D, and radius AK, describe arcs 1, 2,3, 4. By connecting these 
points of intersection a circumscribed square will be obtained. With 
the centers R, 8S, V, IT, and radius RK, describe ares 5, 6, 7, 8 to obtain 
the points G, H, L, N, O, P, E, F, which being connected will com- 
plete the circumscribed octagon. 

DRAFTSMAN’S METHOD. With 45° triangle and T square draw 
tangents FE and LN, GH and PO. At 45° draw tangents FG, ON, 
HL and EP, completing the octagon. 


71. On a given side, AB, to construct a regular polygon having any 
number of sides. Fig. 80. With AB as a radius describe the semi- 
circle D2B and divide it into as many parts as the polygon has sides; 
in this case five. Beginning with the second division from the left 
draw radial lines, A2, A3, A+. A2 will be one side of the polygon. 
Bisect sides AB and A2 to obtain center of circumscribing circle. The intersection of this 
circle with the radial lines A2, A3, and A4, will determine the vertices of the polygon. 





48 








PROBLEMS 


72. Within an equilateral triangle, ABC, to draw three equal circles 
tangent to each other and one side of the triangle. Tig. 81. Bisect the 
angles A, Band C. Bisect the angle DCA.  E is the center of one of 
the required circles. With center K and radius KE describe are EFG. 
F and G will be the remaining centers. From these centers with radius 
EL describe the required circles. 


73. Within an equilateral triangle, ABC, to draw three equal circles 
tangent to each other and two sides of the triangle. I[‘ig.82. Bisect the 
angles A, Band C. Bisect angle DCA. E will be the center of one 
of the circles. With K as center and radius KE, describe are EFG to 
obtain the remaining centers. Draw circles tangent to the sides AB, 
BC and AC.* 


74. Within an equilateral triangle, ABC, to draw six equal circles 
tangent to each other and the sides of the triangle. Tig. 83. Bisect the 
angles and obtain E as in Art. 72. Through E draw HN parallel to 
AC. Draw HM parallel to AB, and MN parallel to BC. With E, H, 
F, M, G and N as centers, and with radius EL, describe the required 
circles. 


75. Within a given circle, ACE, to draw three equal circles tangent 
to each other and the given circle. Fig. 84. Divide the circle into six 


* Instead of bisecting the angle DCA, to obtain one of the centers, describe a semicircum- 
ference, AFC, on one of the sides, as AC, thus obtaining the center F. 


GEOMETRICAL PROBLEMS 


equal parts by diameters AD, BE, CF. Produce AD indefinitely, 
and from E draw the tangent EG. Bisect KGE. With K as center, 
and radius HK, describe are HLM, and with radius HE, from centers 
H, L and M, describe the required circles. 


76. Within a given circle to draw any number of equal circles 
tangent to each other and the given circle. Fig. 85. Divide the circle 
by diameters into twice as many equal parts as circles required; in 
this case eight. Suppose the center of one of these circles to lie on 
AK; then the circle must be tangent to both FK and KB. Draw 
tangent at A intersecting KB. Since the required circle must be 
tangent to the given circle at A, it will also be tangent to AB, and as 
it must lie in the angles FKB and ABK, its center must be at D, the 
intersection of their bisectors. With center K draw circle through D; 
its intersection with EK, CK, etc., will determine the required centers. 
Irom these centers describe the required circles with radius AD. 


77. About a given circle to circumscribe any number of equal 
circles tangent to each other and the given circle. Fig. 86. Divide 
the circle by diameters into twice as many equal parts as circles 
required; in this case six. From A, the extremity of any diameter, 
draw tangent AB. Produce KB making BC equal to AB. At C, 
perpendicular to BC, draw CD intersecting AK produced, at D. 
This will be the center and AD the radius of one of the required 
circles. With center K and radius DK obtain other centers. 


49 





SECTIONS 


CONIC 


















































Fig. 88. 





CHAPTER IV 
CONIC SECTIONS 


78. The Cone and Cutting Planes. Figs. 87 and 88 are illustrations of a right cone with 
a circular base cut by planes making several angles with the axis. It is a complete cone in 
that it extends as much above the vertex A as below it, the two parts being known as the 
upper and lower nappe. It is called a right cone because the axis is perpendicular to the base. 
The curves of intersection between the planes and the surfaces of the cone are known as conic 
sections. ‘They are four in number: the CIRCLE, ELLIPSE, PARABOLA and HYPERBOLA. An 
edge view of the planes is illustrated by Fig. 87, which shows the relation they bear to the 
axis AC and an element AB. 

The circle is obtained by a cutting plane, N, perpendicular to the axis. The ellipse is 
obtained by a cutting plane, R, oblique to the axis and making a greater angle with the axis 
than the elements do. ‘The parabola is obtained by a cutting plane, 5, making the same angle 
with the axis as the elements do. ‘The hyperbola is obtained by a cutting plane, T, making a 
smaller angle with the axis than the elements do. All planes cutting hyperbolic curves will 
cut both nappes of the cone. 

In Figs. 87 and 88, if we conceive the plane T as revolving about the line EF as an axis, 
it will cut the cone in hyperbolas from T to S. At 5, parallel to AB, the curve will be a parab- 
ola. From S to N it will cut ellipses, and at N, a circle. The latter is not shown in Fig. 88. 

51 


52 CONIC SECTIONS 


These curves may be obtained in two ways: First, by determining the curve of inter- 
section between the planes and the cone, as in Fig. 88; second, by known data and a knowl- 
edge of the characteristics of the curve. Only the latter is considered in this chapter. 


79. The Ellipse is a curve generated by a point moving in a plane so that the sum of the 
distances from this point to two fixed points shall be constant. If, in Fig. 89, we conceive 
EKF to be a cord fastened at its extremities, E and F, and held taut by a pencil-point at K, it 
may be seen that as motion is given to the point it will be constrained to move in a fixed path 
dependent on the length of the cord. When the pencil-point is at B, one segment of the cord 
will equal BE and the other BF, their sum being the same as KE plus KF, and also equal to 
AB. The fixed points E and F are called the Foct. ‘They he on the longest line that can be 
drawn terminating in the curve of the ellipse. ‘The line is known as the MAJgor Axis, and 
the perpendicular to it at its middle point, also terminating in the ellipse, is the Mrvor Axis. 
Their intersection is called the center of the ellipse, and lines drawn through this point and 
terminating in the ellipse are known as diameters. When two such diameters are so related 
that a tangent to the ellipse at the extremity of one is parallel to the second, they are called 
CoNJUGATE DIAMETERS. KL and MN are two such diameters. 

In order to construct an ellipse it is generally necessary that either of the following be 
given: The major and minor axes; either axis and the foci; two conjugate diameters; a 
chord and axis. 


80. Ellipse. First Method. Fig. 89. By definition it may be seen that a series of points 
must be so chosen that the sum of the distances from either of them to the foci must equal the 
major axis. Thus, HE+ HF must equal CE+CF, or KF + KE, each being equal to AB. 





THE ELLIPSE 3 


If the major axis and the foci be given to draw the curve, points may be determined as. fol- 
lows: From E, with any radius greater than AE and less than EB, describe an arc. From F, 
with a radius equal to the difference between the major axis and the first radius, describe a 
second are cutting the first. The points of intersection of these arcs will be points, the sum of 
whose distances from the foci will equal the 
major axis, and therefore points of an ellipse. 
Similarly find as many points as may be neces- 
sary to enable the curve to be drawn free-hand. 
Lightly pencil a line through these points. 
For inking see Art. 15, page 11. 

Having given the major and minor axes, 
we can find the foci by describing, from C as a 
center, an are with a radius equal to one-half 
the major axis. The points of intersection 
with the major axis will be the foci; and this 
must be so since the sum of these distances is 
equal to the major axis; and the point C being Fig. 89. ELLIPSE FIRST METHOD 
midway between A and B the two lines CE and 
CF must be equal. Again, if the major axis and foci are given, with a radius equal to one- 
half this axis describe arcs from the foci cutting the perpendicular drawn at the middle point 
of the major axis and thus obtain the minor axis. Having the two axes proceed as before. 

A tangent to an ellipse may be drawn at any point, K, by connecting this point with the 
foci, and bisecting the exterior angle SKE. KT will be the required TANGENT. 





b4 





CONIC SECTIONS 


81. Ellipse. Second Method. Fig. 90. Let AB 
and CD be the major and minor axes of an ellipse. 
Lay off on a piece of paper having a clean-cut edge the 
distance RT equal to one-half the major axis, and RS 
equal to one-half the minor axis. If point T be placed 
upon the minor axis and point S upon the major axis, 
and the paper constrained to move always under these 
conditions, the point R will describe an ellipse. Points 
may be laid off on the drawing to correspond with the 
different positions of R, and through these the required 
ellipse will be drawn. ‘This is an excellent method, 
as construction lines are not required. It is known as 
the method by trammels, since an instrument called the 
elliptographic trammel is constructed on this principle. 


82. Ellipse. Third Method. Fig. 91. Having the 
major axis AB and the minor axis CD, describe circles 
on these as diameters. Draw any radial line, as MG. 
From its intersection with the outer circle draw MO 
perpendicular to the major axis, and from its intersection 
with the inner circle draw NO perpendicular to the 
minor axis. The intersection of these lines at O will 
be a point in the ellipse. Similarly obtain other points. 


THE ELLIPSE 


A tangent at the point O may be obtained by 
drawing a tangent to the outer circle at M and from 
its intersection with the major axis at B, drawing the 
required tangent through O. 


83. Ellipse. Fourth Method. Figs. 92, 93, 94. 
This is a very general method and may be used when 
we have given either the major and minor axes, one of 
the axes and a chord of the ellipse, or any two con- 
jugate diameters. 

CAsE 1. Fig. 92. Having given the major and 
minor axes. From the extremity of the major axis, 
draw B6 parallel and equal to half the minor axis; di- 
vide it into any number of equal parts; in this case 
six. Divide BG into the same number of equal parts. 
Through points 1, 2, 3, etc., on B6, draw lines to ex- 
tremity C of the minor axis. From D, the other ex- 
tremity of the minor axis, draw lines through points 
1, 2, 3, etc., on BG, intersecting the above lines in 
points which will lie in the required ellipse. Construct 
the remainder of the ellipse in the same manner. 

CAsE2. Fig. 93. Having given an axis CD and chord FH. From F draw F4 parallel 
to CD; divide it’into any number of equal parts; in this case four. Divide the half chord FE 





Fig. 93. Cc ELLIPSE eee METHOD 


56 3 CONIC SECTIONS 


into the saine number of equal parts; through these 
points and extremities of given axis draw intersecting 
lines as before, thereby obtaining the elliptical are FD. 
Construct opposite side in the same manner. 

Case 38. Having given the conjugate diameters 
AB and CD, Fig. 94. From A and B draw lines A6 
and B6 parallel to the diameter CD and equal to CG. 
Divide these into any number of equal parts, and, hav- 
ing divided BG and AG into the same number of equal 
5 parts, draw lines from these points to the extremities 
Fig. 94. 00 es" Of diameter CD. .'The intersection of these lines with 

Cc Oo 


| the former, will determine points in the eilipse. In like 

. manner describe the opposite side. : 3 
BS Je 84. Ellipse. Fifth Method. Fig. 95. To describe 

g an approximate ellipse, the major and minor axes being 

| 4 ‘ given. For many purposes in drawing it is sufficiently | : 
accurate to describe the ellipse by means of four circu- 
lar ares of two different radii. ‘The following is one of 

H K several methods: On the minor axis lay off GF and | 


Fig. 95. - ELLIPSE FIFTH MetHoD «6 «equal to the difference between the major and 
| | minor axes. On.the major axis lay off GE and GL 
equal to three-quarters of GF. Connect points F, E, O, L, and produce the lines. ‘With 








M 


THE PARABOLA 3 eye 57 


center E and radius AE describe-are HAM. With center F and radius FD describe are 
KDH. In like manner describe MCR and RBK from centers.O and L. Do not use this 
method when the major axis is moré than twice the minor. 


85. The Parabola is a curve generated by a point moving in a plane so that its distance 
from a fixed point shall be constantly equal to its distance from a given right line. Point F, 
Fig. 96, is the Focus, CD is the given right line called the Drrecrrrx, and AB, a perpendic- 
ular to CD through F, is the Axis. V, the intersection of the axis with the curve, is the 
VERTEX, and by the definition of a parabola it must be equidistant from the focus and directrix. 


86. Parabola. First Method. Fig. 96. Having given the foeus F and the directrix CD. 
Bisect FA to find the vertex V. Through any point on the axis, as L, draw MN parallel to 
the directrix and with radius LA describe arc 1 from focus F as center, intersecting line MN 
at points Mand N. These are points in the parab- 
ola. Similarly obtain other points and draw the 
required curve. <A tangent to: the curve may be 
drawn at any point M by drawing MO parallel to . 
the axis and bisecting the angle OMF. MT is 
the required tangent at the point M. 

Since the angle ‘T’MR is equal to the angle 
TMF, it follows that MR would be the direction 
of a ray of light emitted from the focus F and re- 
flected from the parabola at M. The locomotive ° 
head light is constructed on this principle. 





M 





PARABOLA 


CONIC SECTIONS 


87. Parabola. Second Method. Fig. 97. Having given 
the abscissa VB, and the double ordinate GE. Draw AE and 
CG parallel and equal to VB. Divide AE and BE into the 
same number of equal parts. From the divisions on BE 
draw parallels to the axis and from the divisions on AE draw 
lines converging to the vertex V. The intersection of these 
lines, 1 and 1, 2 and 2, etc. will determine points in the 
required curve. In like manner obtain the opposite side. 


88. Given a parabola to find its axis, focus and directrix. 
Draw any line BG, Fig. 98, cutting the parabola in two 
points. From any point E draw EK parallel to BG. Bi- 
sect BG and EK, and through these points draw LO. From 
any point H draw HM perpendicular to LO. The perpen- 
dicular to this line at its middle point, AN, will be the 
required axis. 

To find the focus draw HV through the vertex, and 
perpendicular to it draw HP. Lay off VF equal to one- 
fourth of NP. Fis the focus. Having laid off AV equal 
to VF, draw the directrix CD perpendicular to the axis. 


89. The Hyperbola is a curve generated by a point 
moving in a plane, so that the difference of the distances from 
this point to two fixed points shall be constant. It will be 


THE HYPERBOLA 


observed that this definition differs from that of the 
ellipse by using the word difference instead of sum. 
The two fixed points F and F’, Fig. 99, are the Foct, 
and the line VV’ is the constant distance called the 
TRANSVERSE Axis. From the definition it will be seen 
that VV! is equal to TF minus TF’, or NF minus NF, 
ete. There will be two branches to the curve. 


90. Hyperbola. First Method. Fig. 99. Having 
given the transverse axis VV! and the foci FEF’ to 
describe an hyperbola. With any radius greater than 
F’V, from centers F and F’ describe arcs 2; from the 
same centers, with radii decreased by VV’, describe arcs 
1 intersecting arcs 2. These points of intersection will 
be points in the required curve. A tangent, ST, may 
be drawn to any point, T, by bisecting the angle FTF’. 


91. Hyperbola. Second Method. Fig. 100. Havy- 
ing given the axis VV’, a double ordinate, AD, and BV, 
its distance from the vertex. Draw AC and DE parallel 
and equal to BV. Divide AB and AC into the same 
number of equal parts, and from points on AC draw 
lines converging to the vertex V. From points on AB 
draw lines converging to the vertex V’. The intersection 













iZ 


SEZ 






XY 





D E« 
Fig. {00. HYPERBOLA! SECOND METHOD 


of these lines 1 and 1, 2 and 2, etc. will be points in the required hyperbola. 






EQUILATERAL: 
HYPERBOLA 








Fig. 104, HYPERBOLA 


CONIC SECTIONS 


92. The Equilateral Hyperbola. Fig. 101. If the hyperbola 
is derived from a cone, the elements of which make angles of 45° 
with the axis, the curve is called an equilateral hyperbola. A knowl- 
edge of the construction of this curve is important, as it is the best 
approximation to the expansion curve of steam and is much used in 
connection with indicator diagrams. 

In Fig. 101, A is the vertex and AB, AC the elements of a cone 
from which the curve is derived. V is the vertex of the hyperbola 
and AV is one-half the transverse axis. Draw VD and VE parallel 
to AB and AC respectively. Draw any line AF radiating from A 
and intersecting VD and VE or these lines produced. From the 
points of intersection F and K draw perpendiculars FL and KL. 
The intersection of these perpendiculars, L, will be a point in the 
hyperbola. It will be observed that the curve on the right of V is 
approaching AB, and on the left it is approaching AC; but it will 
meet and be tangent to these lines only at infinity. Lines having 
this relation to an hyperbola are called the ASYMPTOTES of the curve. 
The asymptotes of the hyperbola illustrated in Fig.99 are AB and AC. 


93. Figs. 102, 103, 104, illustrate the relation between the 
ellipse, parabola, and hyperbola, when constructed by the method 
which is common toall. In each case the height of the curve is the 
same and they have equal chords. 


CHAPTER V 


ORTHOGRAPHIC PROJECTION 


94. THE previous chapters have treated of the penmanship and notation, or the alphabet 
and vocabulary of graphic language. The present chapter will discuss the construction of this 


language, and properly may be called the grammar of 
Graphics. These principles with their adaptation to 
practice by the introduction of various idiomatic con- 
struction, constitute a universal language known as Tech- 
nical Drawing or Graphics. 

The system of representation usually employed is 
ORTHOGRAPHIC PROJECTION, or PROJECTION, as it is 
termed commonly. It is the art of delineating an object 
on two or more planes suitably chosen, and generally at 
right angles to each other, so as to represent exactly the 





a 
NY 
cs) ne 
we 


VERTICAL 
PLANE 


wi 
z 
< 
a 
a 
w 
4 
w 
° 
aw 
a et] 
ey 
°o 


Fig. 105, 


form and dimensions of its lines and surfaces and their relation to each other. 


95. Projection. Suppose that it is required to make the projections of a pyramid having 
a rectangular base, Fig. 105. Conceive the object as surrounded by transparent planes called 
PLANES OF PROJECTION or COORDINATE PLANES and shown in perspective by GLKC, GLBA 


61 


62 ORTHOGRAPHIC PROJECTION 


and GCDA. We may have three representations of the object; one by looking through the 
front or vertical plane, one through the top or horizontal plane, and one through the side or pro- 
file plane of projection. ‘These representations will be the correct projections of the object, if 
we imagine the eye as being directly opposite all of its points at one time, so that the rays of light 
from the points to the eye be perpendicular to the planes of projection; or we may conceive 
the eye as at an infinite distance from the plane, in which case all the rays will be parallel. 
Therefore, if from each point of an object perpendiculars be drawn to the planes of projection, the 
intersection of these perpendiculars with the planes will be the required projections of the points, 
and the lines joining these points will be the projections of the lines and surfaces of the object. 

Thus point 1, Fig. 106, is projected on to the vertical plane GLKC at the point 1’, on to 
the horizontal plane GLBA at 1" and on to the profile plane GADC at 1”. In like manner 
the points 2, 3, 4 and 5 are projected on to the three planes. The small letters Y, ¥ and f, 
above and to the right of the numbers, indicate the planes upon which the points lie, and when 
these letters are not affixed it signifies that the point or line itself is meant and not its pro- 
jection. Since points 1 and 3 are projected on to the vertical plane by the same perpendicular, 
it follows that they will have a common point for their projection, this being designated as the 
projection of two points by the figures 13, the first figure indicating the point nearer the 
plane of projection. Observe similar cases on the profile plane. | 

In order to represent these planes of projection upon a plane surface, as a sheet of drawing 
paper, it is necessary to revolve two of them into the plane of the other one, as in Fig. 106. 
The horizontal plane is revolved about GL into the position GLB’A’’, and the profile plane is 
revolved about GC into the position GCD’A’. By this means we have obtained on a plane 
surface three representations of the object as they would appear on planes at right angles to 


ORTHOGRAPHIC PROJECTION 635 


each other. A good conception of the relation of the planes may be obtained by cutting a 

piece of paper in the form shown by B’A!’GA!D/KB' and then folding it on the lines GL and GC. 
Observe that the top view is always above the front view, and the side view to the right 

or left of the front view according as it may be — ag’ a” 

a view of the right or left side of the object. 

That representation which appears on the 
top plane of projection is called the Tor VIEw, 
HORIZONTAL PROJECTION or PLAN. That on 
the front plane is called the Front VIEW, 
VERTICAL PROJECTION or FRONT ELEVATION. 
That on the side plane is called the SIDE VIEw, 
PROFILE PROJECTION or SIDE ELEVATION. 
‘The view takes its name from the plane on which 
the representation is made and not from the face 
of the object represented. The first of these 
names is most consistent with general practice. k . 
The second forms are used in treatises on De- Fig. 106. 
scriptive Geometry. These terms are employed also in speaking of the codrdinate planes, which 
for brevity are known as H, V and P. The third forms are used in architectural work. 

GL, the line of intersection between the H and V planes, is called the GrounpD LINE. 
GA, the intersection between the profile plane and H, is the horizontal trace of the profile plane 
and is indicated as the Htr. of P. GC, the intersection between the profile plane and V, is 
the vertical trace of the profile plane and is indicated as the V tr. of P. 





64 ORTHOGRAPHIC PROJECTION 


From the foregoing the following laws are established: The front and top views of any . 
point lie in the same vertical line: The front and side views of any point lie in the same horizontal 
line: The top and side views of any point lie equally distant from the ground line and the 
tr S0pee:: 


96. To determine the projections of an object. Suppose it is required to obtain three 
views of a rectangular pyramid, as shown in Fig. 107, having given its dimensions. First draw 
the ground line and traces of P. The view included within the angle LGC will be the front 
view; that within the angle A'’’GL the top view; and that within the angle A’GC the side 
view. It is not necessary to limit these planes by drawing boundary lines, as in Fig. 106, since 
the planes are supposed to be indefinite in extent. In general, draw that view first about which 
most is known. In this problem the views will be drawn in the following order: front, side, 
and top view. At some convenient distance below GL and to the left of GC draw the line 
1V2V of the length required to represent the base of the pyramid. At its middle point erect a 
perpendicular equal to the required height and connect points 1%, 5Y and 2Y. This will com- 
plete the front view. Since the front and side views of any point lie in the same horizontal, 
it follows that the side view of the vertex, point 5, must lie in the horizontal line drawn through 
the point 5Y and at some convenient distance to the right of GC, as 5’. Points of the base 
will be similarly projected, and, since the pyramid is symmetrical, the points 2?, 1°, and 4°, 
3°, will be equally distant from the vertical center line drawn through 5°, the line 2?4? being 
made of the length required to represent the depth of the pyramid. 

To project points from front and side views to the top view it is necessary to observe first, 
that the front and top views of any point le in the same vertical line; second, that the top 


ORTHOGRAPHIC PROJECTION 65 


and side views of any point lie equally distant from GL and V tr. of P. Therefore to project 
any point, as 5, into the top view, draw a perpendicular to GL through 5’ and the point will lie 
in this line; but its position may not be chosen as 
before, since two projections of a point being given 
the third is fixed, and the distance of this point 
from the front plane of projection is determined 
from the side view. Draw a perpendicular through 
the point 5° until it intersects Htr. of P at E, and 
remembering that the lines GA’’ and GA’ are one 
in space, revolve the point E by describing the are 







| lev 
EE’ from the center at G. From E’ draw the hori- | woe vide 
zontal projecting line E’5", which will determine ! “ee thont view Ese el i 
the top view of the point by its intersection with 3 ae 
5Y5". In like manner the points of the base may | | | 
be found, and, since the vertex is connected with a 


VS’ Q’4Y 
the four corners of the base, we shall have the lines 


SEIN, SU2", SN3" and 544%, to represent these edges. 

Observe the following: The projection of a point is always a point. The projection of a line 
is either a point or a line. The projection of a surface is either a line or a surface. ‘To illustrate: 
The point 1 of the pyramid is projected as a point on each of the planes as shown at 1", 1%, 1°. 
The projection of the line 12 is a line on the front and top planes and a point on the side 
plane. The surface 1 5 2 is projected on the front and top planes at 1¥5Y2Y and 1"5"2", but 
on the side plane by the line 1°5?. 


66 ORTHOGRAPHIC PROJECTION 


Observe that the ground line is the horizontal projection of the vertical codrdinate plane, 
and the vertical projection of the horizontal codrdinate plane. In like manner the traces 
of P are seen to be the projections of one or the other of the codrdinate planes. 


| 
Bea 


a 


ae 


AA 


Fig. i08. 


In actual practice the projection lines are seldom 
drawn from one view to another, only such portion being 
pencilled as may be necessary to locate the required 
points. Also, the ground line and profile traces do not 
appear, as it is customary to make the center lines 
serve this purpose wherever it may be possible. Fig. 
108 is a projection of the same pyramid as was illus- 
trated in Fig. 107 with the above-mentioned lines 
omitted. In projecting from the top to the side view, 
or vice versa, the dividers would be used to transfer 
the distances of the points from the ground line and 
traces of P which are now represented by the center lines 
of the figures. 

It is better, however, for the student to make use 
of the ground line and projection lines until he has be- 
come thoroughly familiar with the relation of the object 
to the codrdinate planes. The fact that these three repre- 
sentations of the object are made on planes perpendicular 
to each other must never be absent from his mind. 


OBJECTS OBLIQUE TO THE COORDINATE PLANES 


97. Objects oblique to the planes of projection. In studying the 
representation of an object which is inclined to one or more of the 
planes of projection, it is desirable to observe the changes which take 
place in its appearance when it is revolved in each of three ways: 
first, about an axis perpendicular to H, which is equivalent to turning 
it around on its base; second, by revolving it about an axis perpen- 


dicular to V, which would be inclining it to the right or left; third, - 


by revolving it about an axis perpendicular to P, that is, revolving it 
forward or backward. 

Fig. 109 represents three views of a rectangular pyramid, and it is 
required to represent the pyramid after it shall have been revolved 
about each of the three axes described above. : 

Fig. 110 is the projection of the object after having been revolved 
through 30° about an axis perpendicular to H. In this case the top 
view or horizontal projection will be the same as in Fig. 109, only the 
relation of the lines to the ground line being changed as the base moves 
in a horizontal plane. Both the front and side views are changed, but 
inasmuch as all the points of the object revolve about the required 
axis in planes perpendicular to the axis, the height of the points will 
remain unaltered. ‘Therefore, having copied the top view as de- 
scribed, the front and side views may be projected from it as shown in 
Fig. 110, the height of all points of the object being the same as shown 
inbigs 109. ) 


Fig. 110. 


67 





ORTHOGRAPHIC PROJECTION 


Fig. 111 is the projection of the pyramid after having been 
revolved from the position shown in Fig. 109 through 45° about an 
axis perpendicular to V. This will make no change in the front view 
or V projection, and it may be copied from the front view of Fig. 109,* 
drawing it at the required angle. ‘The top and side views are obtained 
by projecting the points from the front view, observing that as all the 
points revolve about the axis in planes perpendicular to the axis, their 
distance from V will remain the same as in Fig. 109. These dimen- 
sions may be taken from GL or from the parallel plane passing through 
the center. 

Fig. 112 represents the projection of the pyramid after having 
been revolved from the position shown in Fig. 109 through 80° about 
an axis perpendicular to P. In this case the side view will remain 
unchanged from that of Fig. 109, and the projections of the front and 
top views are obtained by noting that all points in revolving about the 
required axis do not change their distances from P, as they do not 
move to the right or the left. 

From the foregoing it will be seen that there is always one view 
and one set of dimensions that will remain unchanged by a revolution 
about any axis. The following statement comprises all that has been 
said concerning the revolution of an object. 

The unchanged view lies on that plane of projection which is perpendicular to the axis of 
revolution, and the unchanged dimensions are parallel to the axis of revolution. 





* Fig. 113 is the same as Fig. 109 on the previous page. 


OBJECTS OBLIQUE TO THE COORDINATE PLANES 69 


In the preceding case all the revolutions Byes ae (is 
were made from one position of the object ; -><}- => a 


but by making successive revolutions it is -~_.__.__. bgp A : 

possible to represent the object in any con- | | 

ceivable position. Having given the projec- | ! 

tions of the pyramid, as in Fig. 113,, the | 

following may be determined : — /\ | <p | i 
Fig. 114 represents the pyramid as re- | | | 


volved from the position -f Fig. 113, 45° to 
the right about an axis perpendicular to V. 
The V projection is unchanged, and all the 
dimensions parallel to the given axis remain 
the same. 

Fig. 116 represents the pyramid as re- 
volved from the position of Fig. 114 back- 
ward through 30° about an axis perpendicular 
to P. The P projection is unchanged, and all 
the dimensions parallel to the given axis re- 
main the same. 

Fig. 115 represents the pyramid as re- 
volved from the position of Fig. 116 through 





| 
Fig. l5, Fig. 116. 


30° about an axis perpendicular to H. The H projection is unchanged and all the dimensions 
parallel to the given axis remain the same. 


70 ORTHOGRAPHIC PROJECTION 


98. Auxiliary planes of projection. Instead of revolving the object to make any required 
angle with one of the planes of projection, as was done in the preceding article, we may revolve 
the plane on which the projection is required. If the object be projected on to this new posi-— 
tion of the plane, the result will be the same as by the previous method. Frequently, too, it is 
desirable to prevent the foreshortening of a surface by projecting it on to a plane to which it 
is parallel, and in such cases it is usually more simple to 
revolve the plane than the object. 

Fig. 117 is a perspective representation of a pyra- 
mid, three planes of projection, and the projection of the 
pyramid on these planes. This differs from Fig. 105, 
page 61, in that the side plane is revolved parallel to the 
right-hand slant face of the pyramid. ‘The size and 
shape of the pyramid are unchanged, and the same nota- 
tion is used. The traces of the planes of projection GL 
and GA remain the same, but GC is changed by reason 
of the revolution of the plane. | 

Fig. 118 illustrates these planes when revolved to 
coincide with V, and the projections of the object as they 
would appear on these planes. It will be observed that the method of projecting is identical 
with that already explained. Having drawn the front and top views, the side view is obtained 
by drawing perpendiculars from the V projection of the points to the trace GC, and determin- 
ing the position of these points on the perpendiculars by projecting from the top view as 
indicated. Thus the side view of point 2 will be at the same distance from GC that the top 





Fig. 117. 


AUXILIARY PLANES et 


view is from GL. The projecting lines may be used, or the measurements may be made with 
the dividers. In practice it is common to omit the traces, the center lines of the figure being 
substituted for them as was suggested on page 66. In the illustration the auxiliary plane is 
drawn parallel to the face 254, and therefore that face will not be foreshortened in this 
view. ‘The lines will be seen in their true length, 
and the triangle in its true shape and area. 


B 


This method is used in problems relating to 
the development of surfaces and wherever a 
view of a special face is required without fore- 
shortening. In such cases it is customary to 
project only those lines which lie in a plane 
parallel to the auxilary plane. 


99. Views omitted by the use of auxiliary 
planes. It is frequently possible to lessen the 
labor involved in the representation of an object 
oblique to the codrdinate planes by obtaining a 
view of some special face of the object having x CG 
dimensions necessary to the making of the other Fig. (18. 
views. In such a case an auxiliary plane parallel to the important face would be used. 
Suppose that it is required to draw an octagonal prism having its edges parallel to the 
vertical codrdinate plane, and the base making an angle of 30° with the horizontal plane. One 
of the long diameters of the base is parallel to the vertical plane. 





72 


ORTHOGRAPHIC PROJECTION 


Fig. 119 illustrates two views of the prism in the required position. ‘These might be ob- 


tained by drawing the prism as resting on its base with its edges and a long diameter of the 





bases parallel to V, and having obtained the front and top views 
it would then be revolved about an axis perpendicular to V until 
the bases made the required angle with H; but by the following 
method the first of these operations will not be required. As the 
front view is unchanged by a revolution about an axis perpen- 
dicular to V, the upper and lower bases MYNY and AYEY may be 
drawn, if the diameter of the circumscribing circle of the base be 
given and the height of the prism be known. ‘To determine the 
projection of the other edges the points BYCYDY must be obtained. 
Suppose the base to be projected on to an auxiliary plane parallel 
with itself, and this plane revolved about GC to coincide with V. 
The representation on this plane will be the same as the top view 
of the prism before having been revolved into the position shown, 
one half of which is projected by AYB/C’D’/EY. By counter revo- 
lution B’ will be projected at BY, C’ at CY, and D’ at DY. The 
edges being drawn perpendicular to the bases will complete the 
view. ‘To draw the top view it will be necessary to obtain the 
distances BYL" and C¥K#. These can be measured from the view 


on the auxiliary plane, as they are double the distances B/BY and C’CY. Having located the 
corners, complete the lower base and draw the upper base parallel to it, the foreshortened edge 
AM being projected from the front view. 


AUXILIARY PLANES 1 


100. The projections of a circle oblique to the coordinate planes. It is required to find the 
front and top views of a circle lying in a plane perpendicular to V and making an angle of 30° 
with H; also to determine its projections after having been revolved from this position 30° 
about an axis perpendicular to H. The two views of the first position, Fig. 120, are to be 
obtained in a manner similar to that described for the base of the prism in the preceding 
article. Draw the diameter AYBY at the required angle of 30° and this will be the front view 
of the circle. We may conceive the circle as projected 
on to an auxiliary plane parallel to itself, or suppose 
the circle to be revolved about the diameter AY BY until 
it is parallel to V, as at AYE’C’BY. The top view of 
any point, as E™, will lie in the vertical drawn from 
EY and at a distance from the diameter A"B" equal to 
E/EY. Obtain other points in the same manner, ob- 
serving that points on the opposite side of the diameter 
A¥BF may be laid off at the same time. 

Fig. 121 illustrates two projections of this surface 
after having been revolved through 30° about an axis 
perpendicular to H. The top view being unchanged 
and the heights of the points remaining the same, the 





front view is readily obtained as indicated. It is very 
important that the points be obtained consecutively, 
and not by drawing the vertical and horizontal projecting lines and finally attempting to 
locate the required points at the intersection of these lines. 


Fig. (20. Fig. (21. 


B" 
ee 
AY 








ORTHOGRAPHIC PROJECTION 


101. The projection and true length of lines. FIRST Mrernop. Let 
A®#B# and AYBY, Fig. 122, represent two views of a line inclined to the 
codrdinate planes H and V. It is required to find its length and the 
angles which it makes with the H and V planes of projection. 

Since the line is inclined to both planes, its projections on them will 
be foreshortened views of the line. In order that the line shall be parallel 
to one of the planes, and thus seen in its true length on that plane, it will 
be necessary to revolve it about an axis perpendicular to one of the planes. 
according to the principles already established for the revolution of an 
object. Art. 97, page 67. It may be revolved about an axis perpendicu- 
lar to H until it becomes parallel to V, or about an axis perpendicular to 
V until parallel toH. Fig. 123 represents the line revolved parallel to V, 
the revolved position being shown by AYB’. The horizontal projection 
of the line, which in the revolution about a vertical axis we know to be 
unchanged, is first drawn parallel with the ground line, in which position 
the line will be parallel to V. The height being unchanged the vertical 
projection may next be obtained, and the representation will be that of a 
line parallel to V and therefore seen in its true length on that plane. 
This view also shows the correct angle, R, which the line makes with the 
horizontal plane, since its position relative to that plane is unchanged 
by the revolution. 

Similarly the line may be revolved about an axis perpendicular to V 
until it is parallel to H, as in Fig. 124, when its true length may be 


TRUE LENGTH OF LINES 


measured on the top view. This view also gives the angle, R, which the 
line makes with V. | 

SECOND MeruHop. If the quadrilateral formed by a line, the pro- 
jecting lines of its extremities, and its projection, be revolved about the 
latter until the surface coincides with the plane of projection, the revolved 
position of the line itself will exactly represent the length of the line, and 
the angle which it makes with the plane of projection will be shown on 
the plane into which it has been revolved. 

In Fig. 125, let A? B® be the horizontal projection of a line, and AYC, 
BYD, the vertical projection of the projection lines of its extremities. 
Since these last-named lines are seen in their true length in this view, we 
have three sides of a quadrilateral, and the fourth will be the true length 
of the line. To obtain this result draw A™A! and B"™B!’ equal to AYC 
and BYD, respectively, and perpendicular to A®B™. The points A’, B’ will 
be the extremities of the revolved position of the required line, and the 
angle which A’B!’ makes with A™B#® is the angle which the given line 
makes with H. ‘The plane in which this quadrilateral lies is known as 
the horizontal projecting plane of the line, and A™B# is called its horizon- 
tal trace, it being the line in which the projecting plane intersects H. 

Similarly, in Fig. 126, the vertical projecting plane of the line has 
been used and the revolution made into V about its vertical trace AYBY, 
which is the vertical projection of the given line. The true length of the 





Fig. 126. 


line is A’B’, and the angle between A’B’ and AYBY is the angle which the given line makes with V. 


76 ORTHOGRAPHIC PROJECTION 


102. Projection of rectangular surface by auxiliary plane. An application of the fore- 
going principles is illustrated by Fig. 127. This represents a rectangular surface in four 
positions, its long sides making angles of 0°, 30°, 60° and 90° with a vertical plane, and the 
short sides parallel to V, and making angles of 55° with a horizontal plane. ‘The surface 
might first have been drawn with its long sides horizontal, and making the required angle 
with the vertical plane, after which it would have been revolved about an axis perpendicular 
to V until the short sides made the required angle with the horizontal plane. This would 
have necessitated the drawing of two views, which may be avoided by 
the use of the following method. 

Suppose the required surface to be drawn in the plane of V as 
shown by AYBYDYCY, the short side making the angle of 55° with H. 
Next conceive the surface as revolved about one of the short sides as 
an axis until the required angles with V are obtained. AYCY is the 
vertical projection of the side which is used as an axis. Draw AVYB! 
equal to AYBY, and making an angle of 30° with it. This will represent 
the position of one side of the rectangle when revolved about the vertical 
trace of its vertical projecting plane, precisely as was done in Fig. 126. 
Fig. 127. By counter revolution about this trace the hne AYB’ is brought into the 

required position, as shown by AYB,’. The horizontal projection of 
this line may now be obtained, since one of its extremities is in V and the other is 
determined by B/B,Y and will be laid off at B,". As the short and long sides are parallel 
each to each, their projections will be parallel and the projections of the surface may be 
completed. ‘The other positions are similarly determined. 





GENERAL PRINCIPLES T7 


103. Rules governing the relation of lines and surfaces to the H and V coordinate planes. 
If a line is perpendicular to either codrdinate plane, its projection on that plane will be a point, 
and its other projection will be perpendicular to the ground line. 

If a line is parallel to one codrdinate plane and oblique to the other, its projection on the 
first plane will be parallel to the line itself and equal to the true length of the line. Its other 
projection will be parallel to the ground line and shorter than the actual line. 

If a line is parallel to both codrdinate planes, both of its projections will be parallel to the 
ground line, and their length equal to that of the actual line. 

Parallel lines will be parallel in projection. 

If a surface is parallel to either codrdinate plane, its projection on that plane will be the 
true shape of the surface, and its projection on the other plane will be a line parallel te the 
ground line, 


CHAPTER VI 
ISOMETRIC AND OBLIQUE PROJECTION 


104. Iv is frequently necessary to produce a pictorial effect in mechanical drawings while 
preserving the relative proportions of parts so that they may be drawn to a scale and measure- 
ments taken from them. ‘Two methods are in general use for accomplshing this end. The 
first is known as AXONOMETRIC PROJECTION, of which isometric projection is a special case ; 
the second is OBLIQUE PROJECTION. Both methods employ but one plane of projection. 


105. Axonometric Projection. Ly this system the object is so related to the plane of pro- 
jection that the foreshortened length of three mutually perpendicular axes bear known relations 
to each other; and, by the use of special scales, lines parallel to these axes may be measured. 
Figs. 128 and 129 illustrate two cases of axonometric projection, the object represented being 
acube. In Fig. 128 all of the edges are equally foreshortened. In Fig. 129 the axes or edges, 
CG and CD, are foreshortened a lke amount, while CB is foreshortened twice that of the 
other two. The angles of these axes being known and the reduced scale due to the fore- 
shortening being given, the making of the projection is very simple. 


106. Oblique Projection. Fig. 130 is an oblique projection of the same cube. ‘This 
differs from the preceding in that one face of the object is parallel to the plane of projection, 
| 78 


ISOMETRIC AXES 


and the projection is made by oblique lines instead of perpendiculars 
as in orthographic projection. See Art. 117, page 86. 


107. Isometric Projection. Fig. 128. This is a special case of 
Axonometric Projection, in which the mutually perpendicular axes 
make equal angles with the plane of projection, and are therefore 
equally foreshortened. 

To obtain this representation by orthographic projection, con- 
ceive the cube as resting on one face with its vertical faces making 
angles of 45° with V. Next suppose the cube .to be revolved toward 
V about an axis parallel to the ground line until the diagonal of the 
cube drawn through C is horizontal. Then its projection will be 
isometric, all of the edges being of equal length and making angles 
of 90° or 30° with a horizontal. 


108. The Isometric Axes. In Fig. 128 the lines CD, CB and CG 
are called isometric axes, and lines parallel to them are known as 
isometric lines. Planes including isometric lines are known as iso- 
metric planes. It is evident that only isometric lines may be measured, 
since they alone are equally foreshortened. ‘Thus the isometric of the 
diagonals of the squares, AC and DB, are of unequal length, although 
in the original cube we know them to be equal. Likewise it is not 
possible to measure directly the angle between lines on an isometric 
<lrawing. 


19 


@ Fig. 128. 
A 


G : 
Fig. 129. 


Fig. 130. 


B 


80 ISOMETRIC AND OBLIQUE PROJECTION 


109. The Isometric Scale. If a scale should be used for making isometric projections it 
would be .814 of full size and constructed as follows: Fig. 135. Lay off a full size scale on 
a 45° line and drop verticals from these divisions on to a 50° line. ‘The latter will be the iso- 
metric scale and may be used for measuring all lnes parallel to the mutually perpendicular 
axes. ‘The inconvenience due to the use of such a scale has led to the adoption of the full 
scale for all isometric representations ; and although this results in an enlarged representation 
of the object, it is seldom noticeable. ‘The term isometric drawing is usually employed to’ 
denote this enlarged representation. 


110. To make the isometric drawing of a cube. Fig. 135. From the point C draw lines 
CB and CD at angles of 30° and equal to the required length of the edges. Draw the vertical 
CG of same length. As each edge of the cube is parallel to one or the other of these isometric 
lines, the drawing may be completed as shown. It is customary to omit the representation of 
invisible lines to avoid confusion. In shading an isometric drawing it is customary to draw 
shade lines for the division between hght and dark surfaces, the direction of the light being 
that of the diagonal DF. 


111. Non-isometric Lines. Fig. 131 is the drawing of a pentagon and Fig. 132 is an iso- 
metric drawing of it. As but one of the lines of the pentagon can be isometric, the construction 
is as follows: DE being chosen as an isometric line or axis, a second axis, YY, is drawn 
isometrically perpendicular to it. Points C and F lie on this axis, and their position is readily 
determined, since measurements may be made on this line: As line AB, drawn through C, is 
an isometric line, points A and B may be located at their proper distances to the right and left 
of the axis YY. 


ISOMETRIC OF A CIRCLE 8] 


112. To make the isometric drawing of a 
circle. Fig. 135. Suppose the circle to be 
inscribed in the square D’C’G’H’. As the iso- 
metric drawing of every circle is an ellipse, it is 
only necessary to obtain the major and minor axes 
to enable the curve to be described by the method 
of trammels, Art. 81, page 54. These axes le on 
the diagonals DG and HC, and their extremi- 
ties may be determined as follows: The diago- 
nals being non-isometric lines, the distance DK 
cannot be measured, and the point K must be 
determined by measurements parallel to the iso- 
metric axes, as PK and OK, which equal P’K’ and 
O’K’. Through K, an extremity of the major 
axis, draw KN parallel to DC; its intersection 
with the second diagonal at N will determine one 
extremity of the minor axis. Obtain M and N, 
and draw the ellipse by the method of trammels. 

A much used approximate method is as fol- 
lows: Bisect the edges of the upper face, Fig. 
135, and connect these points S and T, R and V, ; G 
with the vertices C and A. From their intersec- Aya VERE 
tion, Y and Z, describe arcs RS and TV, and from centers C and A describe arcs ST and VR. 





82 ISOMETRIC AND OBLIQUE PROJECTION 


113. The measurement of angles lying in isometric planes. 
Suppose it is required to make the isometric drawing of the 
lines B/E’, C/E’ and D/E’, Fig. 186, making angles with A’E! 
of 15°, 80° and 45°, respectively. Since the isometric angles 
cannot be measured by means of the included arcs, from any 
point on E’A!’ draw a perpendicular, F/K’, and thus obtain two 
lines which will be parallel to the isometric axes and on which 
the angles may be measured. Having drawn EA, Fig. 187, at 
an angle of 30°, lay off EF equal to E’F’ and draw the vertical 
FK. On this lay off FG, FH and FK equal to F’G’', F’H’ and 
EF’ kK’, and draw the required lines EB, ECand ED. Although 
the angles B/E’A’, C’/E’B! and D’E’C! are equal, they will not 
be so in isometric. 


114. To make an isometric drawing of an oblique timber 
framed into a horizontal timber. Tig. 138 illustrates a side 
view of the timbers, and Fig. 159 the isometric drawing. 
Having made the isometric of the lower piece, the cut for the 
oblique timber should be shown. The edges of this cut being 
non-isometric, they must be obtained by locating the points C, 
D and E, as in Fig. 188. Suppose the required pitch of the 
oblique timber to be two-thirds, that is, two vertical units for 
every three horizontal units. From C, Fig. 139, lay off on AB 





SUGGESTIONS FOR SPECIAL CASES 


any three units and erect a perpendicular equal to two of the units. 
The point found will determine the pitch of the oblique timber. 
Although H’L’ is perpendicular to H’C’, Fig. 138, it is not possible 
to make this measurement directly on the isometric drawing, as it is 
a non-isometric line; but if a perpendicular, H’K’, be drawn from 
H’, this distance may be laid off from H, Fig. 139, thereby deter- 
mining a point, K, on the lower side of the timber through which a 
parallel to HC may be drawn. Finally, the point L may be 
obtained by laying off MN equal to M’N! and erecting a perpen- 
dicular to intersect the lower edge of the timber. 


115. Suggestions for special cases. In general, make the rep- 
resentation of rounded surfaces, fillets and small circles by the 
approximate method, Art. 112, page 81, but do not sketch them 
free-hand. In Fig. 140, the determination of the radii R and R’ 
will enable the seven visible rounded corners to be easily drawn. 

In the representation of a column or shaft, as in Fig. 141, the 
approximate method is sufficiently accurate, since it is not possible 
to measure the diameter directly, and the circumscribed square of 
the approximate ellipse is the same as that of the true ellipse. 

If the isometric drawing of a circle inscribed in a hexagon is de- 
sired, it will be necessary to draw the true ellipse, as in Fig. 142, other- 
wise the ellipse would not be tangent to the sides of the hexagon. 








84 ISOMETRIC AND OBLIQUE PROJECTION 


In representing incomplete work, such as 
the studs, sill and joists in Fig. 143, it is neces- 
sary that the studs be drawn of equal lengths 
in order that they may appear to lie in the same 
vertical plane. It is desirable, also, that all cut 
sections he in isometric planes. 

In laying out irregular curves, draw a 
series of ordinates perpendicular to an edge 
which is to be parallel to an isometric axis, as 
in Fig. 144. This illustrates the true section 
of a molding and its isometric projection. 





116. A useful case of axonometric projection 
is illustrated by Fig. 145, in which the effect 
produced is more nearly that of perspective. 
While this representation involves a little more 
labor than the isometric drawing, it avoids 
much of the distortion which characterizes the 
latter. The system is particularly well adapted 
to illustrating groups of buildings where per- 
spective cannot be used to advantage. It is 
also suitable for Patent-Office drawings. 

The theory upon which the representation is made is as follows: A cube is revolved 





AXONOMETRIC PROJECTION 


into such a position that two of the axes or edges, as CG and CB, 
are equally inclined to the plane of projection, while the third axis, 
CD, is foreshortened one-half that of CG and CB. The angles 
which the edges make with a horizontal line are 7.2°, 41.4° and 90°. 
This involves the use of two special triangles, one of which must 
have a right angle to enable the vertical lines to be drawn. Two 
scales are used; a full scale for dimensions parallel to CB and CG; 
and a half scale for those parallel to CD. As in isometric, all 
dimensions must be made parallel to one of the three axes. The 
ellipse on the face BCGF may be described by obtaining the major 
and minor axes, as in the case of the isometric ellipse, and with 
centers on these axes describing arcs tangent to the edges. ‘The 
axes of the ellipse on face DCGH do not coincide with the diag- 
onals; but a very close approximation to them may be obtained by 
drawing lines KL and MN respectively perpendicular and parallel 
to CB. In unimportant work the extremities of these axes may be 
estimated by the eye; but if accuracy is required, lines KL and MN 
must be drawn on a square of the given size, and their intersection 
with the circle found as was done with the diagonals in isometric 
projection. A comparison of these methods is shown in the accom- 
panying illustrations. Fig. 146 illustrates a chamfered bolt-head 
drawn by this method, and Fig. 147 represents the same by iso- 
metric projection. 


Fic. 146. 


Fig. 14-7. 





86 





ISOMETRIC AND OBLIQUE PROJECTION 


117. Oblique or Cabinet Projection. If one face of the object 
be parallel to the codrdinate plane and the projecting lines oblique 
to the plane, the effect produced will be that illustrated by Fig. 
148, which is the oblique projection of a cube. The face parallel to 
the plane is unchanged, but lnes perpendicular to the plane are 
projected as oblique lines, the angle of which may vary. If the 
projecting lines make an angle of 45° with the codrdinate plane, 
there will be no foreshortening of lines of the object which are per- 
pendicular to the codrdinate plane, as shown by the illustration, 
and all lines parallel to these may be measured as in isometric. In 
Fig. 148, the lines make angles of 45°, but any other angle might 
have been employed. As in isometric projection, care must be used 
to make all measurements parallel to the three axes, save when 
working on the face parallel to the codrdinate plane. The oblique 
projection of a circle on the front face will be a circle, but on the 
side or top face it will be an ellipse. The axes are obtained from 
the front face as shown by the dotted lines, and other points simi- 
larly determined. 

If the angle of the projecting lines be greater than 45°, the 
lines perpendicular to the front face will be foreshortened, pro- 
ducing an effect analogous to perspective. Fig. 149 illustrates a 
cube drawn by this method, in which the oblique lines are fore- 
shartened one-half. This necessitates the use of two scales, the 


al 


CABINET PROJECTION 87 


45° lines being drawn half size. Circles may be obtained as in the preceding case, but it should 
be observed that the diagonals of the face do not coincide with the axes of the ellipse. This 
form of projection is sometimes distinguished from that first described by calling it CABINET 
PROJECTION. One advantage which oblique projection and its modifications have over isometric 
projection is in the representation of one face without foreshortening. It is well adapted 
to the representation of furniture and cabinet work. 

Fig. 150 illustrates the application of these principles in the representation of a card 
cabinet. 


CHAPTER VII 
THE DEVELOPMENT OF SURFACES 


118. To develop a surface. It is frequently required to illustrate the surfaces of an object 
in such a manner that a pattern being made from it and properly folded or rolled would exactly 
reproduce the object. In order to do this, an outline of each surface must be obtained as it 
would appear on a plane of projection parallel to it, so that there will be no foreshortening of 
the surface. Fig. 151 is the projection of a triangular pyramid, and it is required to produce 
the pattern which if properly folded would make a pyramid like the one in the drawing. ‘This 
operation is called the development of the surface. Since the three slant surfaces of the pyra 
mid are triangles, it is possible to obtain their true area by finding the length of their sides. 
A line is seen in its true length on a plane when it is parallel to that plane. Art. 101, page 
74. Thus, the line DC may be measured on the front view because the line of the pyramid 
which it represents is parallel to that plane, and we know that it is parallel to the plane because 
the top view of the line is parallel to the ground line. Neither of the lines DA or DB may be 
measured from the drawing, but, since the base of the pyramid is symmetrical with respect to 
the axis, we know these lines or edges to be of equal length with DC. The only undetermined 
line of the surface BDA is AB, which may be measured on the top view, as it is parallel to 
H. Fig. 152 shows these lines in their proper lengths and relation to each other. Since the 

88 


THE DEVELOPMENT OF A PYRAMID 


surfaces BDA, BDC and CDA are equal, the 
edges of the first having been obtained, the 
others may be copied from it. The bottom 
surface alone remains to be drawn, and this 
being parallel to H is already seen in its true 
size and shape, and may therefore be copied 
directly from the top view. 

Having the length of one of the edges, we 
might have used it for a radius to describe the 
arc BACB, and, by spacing off the bottom edges, 
BA, AC and CB, have attained the same result 
in an easier and more practical manner. 

The surface BDA may be obtained in 
another manner as follows: Since the true size 
of a surface is always to be found on a plane to 
which it is parallel, we have only to draw a 
plane, GC, parallel to this surface and project 
‘on to it, in order to obtain the required surface. 
Art. 98, page 70. This new plane is perpen- 
dicular to the front plane, but not to the other 
planes. GC is its V trace, and, if the plane be 
revolved to coincide with V, the true shape of 
the surface BDA will be obtained, as at B’D/A’. 





Cc 


Fig. 151. 





ear 


D? 


Ci 


89 


90 : THE DEVELOPMENT OF SURFACES 


119. To develop a pyramid when cut by a plane. Fig. 153 illustrates a rectangular 
pyramid which it is required to develop after removing such portion of the top as lies above 
the plane FGHK. The operations are as follows: First, obtain the three views of the object 
before having been cut by the plane: Second, determine the projections of the cut surface 
FGHK, thus showing the pyramid as it would appear with the top portion removed: Third, 
obtain the true shape of the cut surface: Fourth, obtain the development of the entire pyramid, 
disregarding the cutting plane: Finally, determine that portion of the developed surface not. 
removed by the cutting plane, to which must be added the section cut by the plane. As the 
first three operations are sufficiently well indicated by the drawing, we will consider the devel-. 
opment only. None of the inclined edges being parallel to either of the planes of projection, it. 
is necessary to revolve one of these edges until it shall become parallel to a plane, when it will 
be possible to measure it on that plane. Let AE be revolved parallel to V. Since this is a 
revolution about an axis perpendicular to H, the top view of the line will be changed in posi- 
tion, but not in length, and will be shown by A’E®. The front view will then be A!/EY, the 
true length of the line. Since all the inclined edges are of the same length, with radius equal. 
to A’’EY, describe an are on which the chords AB, BC, CD and DA may be drawn, as shown 
in Fig. 154, their lengths being obtained from the top view. ‘The development of the base is 
obtained directly from the top view. 

Finally, having obtained the development of the entire pyramid, it is required to find the 
length of the edges when cut off, and the section made by the cutting plane. Since we have 
found it possible to obtain the true length of the inclined edges, we may in like manner find 
that portion of them included between the base and cutting plane. A/’’EY may be considered 
as the revolved position of any one of these lines, and, since the heights remain unchanged by 


THE DEVELOPMENT :OF A PYRAMID 91 


revolving about an axis perpendicular to H, the 
true length of AK and DH will be A’'N, and 
the length of BF and CG will be A'’0.—It 
is generally better to lay off the distances from 
the apex instead of the base.— The cut surface 
F’G'H'K’ having been found by projecting it on 
to a plane to which it is parallel, it may now be 
copied as a part of the developed surface. In 
like manner the base should be drawn in connec- 
tion with the development so that there may be a 
less number of edges to unite-when the pattern is 
folded to form the required object. 

The edges KF, FG, GH and HK of the de- 
veloped slant faces should be equal to the edges 
of the surface cut by the plane, and projected on 
the auxiliary plane at K/F’, F/G’, G’H' and H’K’. 
As these lines have been obtained independently, 
the comparison of their lengths will serve as an 
excellent test of the accuracy of the drawing. 

The division lines between the faces of the 
prism as shown in the development are represented 
by dashes, because no line really exists until the 
surface has been folded. 





Cc 
Fic. 154. 


92 THE DEVELOPMENT OF SURFACES 


120. The development of surfaces of revolution. A surface may be conceived to be gen- 
erated by the motion of a line. Such a line is called the GENERATRIX, and the different 
positions of the generatrix are called ELEMENTS. The line which may direct or govern the 
motion of the generatrix is called the DIRECTRIX. 

Suppose a right line to have a motion about another right line known as the Axis, from 
which it is always equidistant and to which it is parallel; then will the successive positions of 
this generating line constitute the surface of a cylinder. If one end of the generatrix were 
fixed to the axis and the other free to describe a circular path, the surface generated would be 
acone. These surfaces are called SURFACES OF REVOLUTION, and may be regarded as con- 
sisting of an infinite number of lines or elements which in the first case are parallel to, and 
in the second case intersect, the axis. This conception of the cylinder and cone is necessary 
to the study of the development and intersection of surfaces. 


121. The development of a cylinder. Three views of a cylinder are represented by Fig. 
155, and from these it is required to develop the cylinder. Assume a number of elements, 
and for convenience they should be equidistant. These may be employed to obtain other 
views, sections and development in precisely the same manner as though they were the edges 
of a prism, save that, instead of connecting their extremities by right lines, a curve must be 
drawn through them. It will be observed that the revolved section is an ellipse, the major 
axis of which is equal to AYCY, and the minor axis equal to the diameter of the cylinder. 
From these data the curve might be drawn, but it is better to use this method as a test for 
the ellipse after having obtained the curve by means of the elements. 

To develop the cylinder, Fig. 156, obtain the length of the base DD by determining the 


THE DEVELOPMENT OF A CYLINDER 93 


circumference of a circle having a diameter equal to 
that of the cylinder, and, having laid it off, divide it 
into as many parts as there are elements. ‘The per- 
pendiculars to the base drawn through these points —-, a 
will be the required elements, and their lengths may 
be obtained directly from the front view, since they 
are parallel to the vertical plane. A free-hand curve 
should be carefully pencilled through these points 
and afterwards neatly inked by the aid of compasses 
and curves. 

It is unnecessary to add the base and cut section 
to the development, as was done in the case of pyra- 
mids, since there will be but one point of contact 
between these surfaces. 

If the base of the cylinder had been an ellipse 
instead of a circle, it would have been necessary to 
obtain the development by spacing the length by 
the dividers or spacers. In such cases do not use 
a unit greater than 10° or 15°, and obtain the entire 
length of the base before locating the elements. 











94 THE DEVELOPMENT OF SURFACES 


122. To develop a cone. Fig. 157 illustrates a cone cut by vertical and oblique planes. 
Draw the requisite number of elements by dividing the circle of the base in the top view and 
projecting these points into the front view, as at BY, EY, GY, HY, etc. These are then con- 
nected with the vertex AY. It is now possible to obtain the top view of the section made by 
the cutting plane XY in the same manner as though the cone were a pyramid having these 
elements for its edges. A more accurate method is as follows: Through the front view of 
any point NY, lying on the surface of the cone and also in the cutting plane, draw a horizontal 
line which will represent the front view of.a circle of the cone, and be shown on the top view 
by the fine dotted circle drawn through N™. Since the assumed point must lie on this circle 
as well as on its vertical projecting line, it must lie at their intersection N¥. Of course it is 
necessary to draw only short ares to intersect the projecting lines. In like manner any num- 
ber of points may be obtained and through them the curve described. If the elements of the 
cone have already been drawn, as would be necessary if it is to be developed, the points 
assumed had better be at the intersection of the cutting plane and the elements. This curve 
being an ellipse may be obtained by finding the major and minor axes and on these construct- 
ing the curve; but this method should be used only as a test of the ellipse until the problem 
is thoroughly understood. 

The side view illustrates the true section made by the vertical cutting plane, since it is 
parallel to the profile plane of projection. The top view of this section is a straight line, but 
it is Just as necessary to find the points by projection as in the case of the ellipse, since their 
distances on either side of the axis must be known in order to obtain the side view. Thus the 
top view of point O must lie in the vertical projecting line drawn through OY, and also in a 
circle the diameter of which is equal to VM, therefore at O". From this the side view of the 


THE DEVELOPMENT OF A CONE 95 


point may be obtained, its distance from the axis 
being equal to QUO#. 

Proceed with the development as in the case 
of the pyramid, observing that because all the ele- 
ments are of equal length the development of the 
base will be a circular are of length equal to the 
circumference of base, and radius equal to the length 
of an element. Divide the arc into as many parts 
as there are elements, and proceed to draw the ele- 
ments, after which the true length of the cut portion 
may be found as follows: Having drawn any ele- 
ment, as AG, Fig. 158, it is required to find the 
points P and R. Since the line AG upon which 
they lie is not seen in its true length in any of the 
views, it must be revolved parallel to one of the 
planes. AYBY will represent its position when 
revolved parallel to the front plane. The point PY 
will be seen at P’, and RY at R’, and the lengths 
AYP!’ and P’/R’ may be laid off on the line AG of 
the developed surface. In a similar manner other 
points are found, AB serving as the revolved posi- 
tion for all of the elements, since they are of equal 
length. 





CHAPTER VIII 
THE INTERSECTION OF SURFACES 


123. The intersection of cylinders. Three views of two intersecting cylinders are shown 
in Fig. 159. It is required to determine the curve of their intersection and to develop the 
cylinders. By assuming elements of one cylinder and finding their intersection with the second 
cylinder, points in the desired curve may be obtained. Assume an element of the small cyl- 
inder, the side view of which is A?B’, and determine the H and V projections. The H 
projection, A¥B", is seen to pierce the large cylinder at the point B™, which is the H projection 
of an element of the large cylinder. The V projection of these intersecting elements will be 
CYBY and AYBY. Their point of intersection will be common to both cylinders and, therefore, 
a point in the required curve of intersection. In lke manner obtain a sufficient number of 
points to determine the curve. For convenience in the development of the small cylinder, it 
is desirable to have its elements equidistant. 

In all problems relating to the development of surfaces it is important to determine those 
points of the curve, known as limiting points, at which the direction of curvature changes or 
points of tangency occur. ‘These define the character of the curve and enable it to be drawn by 
the finding of a less number of points. 1°, 2, 3’, 4Y, 5Y and 6 are limiting points of this curve. 

In developing the small cylinder, Fig. 160, open it on the element L4, which will make it 
symmetrical with respect to the center. The method of developing does not differ from that of 

96 


THE INTERSECTION OF SURFACES 


the cylinder in Art. 121, page 92, and 
the length of the elements may be 
taken from the front or top views. 

The development of the large cyl- 
inder, Fig. 161, will be a rectangle 
pierced by a hole which is symmetrical 
with respect to a horizontal center line 
only. Having obtained the develop- 
ment of the cylinder by opening it on 
GH, the element DE will be drawn in 
the center of the surface, and from this 
the other elements may be determined. 
On DE lay off the points 8 and 5 equi- 
distant from the center line and equal 
to that portion of the element cut by 
the small cylinder, as shown on the 
front view. The distance between 
any of these elements, as DE and CN, 
may be found by measuring the circu- 
lar arc D#22C#, which should then be 
laid off to the left of ED, and through 
this point CN may be drawn. Having 
determined the position of an element, 


immediately lay off the amount cut out by the second cylinder as seen on the front view. 


H 





97 


98 THE INTERSECTION OF SURFACES 


124. The use of auxiliary planes. In determining the curve of intersection between two 
surfaces it is customary to use a system of planes which shall cut either circles or straight lines 
from these surfaces. The intersection of these lines will be points of the curve. Let it be 
required to find the intersection of cylinders 1 and 2, Fig. 162. If we imagine them cut by a 
plane, VW, parallel to the axes, the appearance will be as in Fig. 164. Two elements will 
have been cut from each cylinder, and, since they lie in the same plane, their points of intersec- 
tion will be common to both cylinders, and therefore in the required curve. Again, if we 
should employ a plane tangent to cylinder 2, it would cut that cylinder in a single element, and 
the other in two elements, as in Fig. 163. Their intersection at points 7 and 8 would be the 
limiting points of the curve and ordinarily the first to be determined. 

To obtain the elements cut by these planes proceed as follows: Revolve the bases of the 
cylinders, as in Arts. 99 and 100, pages 72 and 73, and assume a cutting plane VW, shown on 
the top view. On the revolved bases lay off LM and NO equal to the distance of the cutting 
plane from the plane of the axes. ‘Through the points M and O draw parallels to the bases, 
and these will indicate the amount cut from each cylinder. Next revolve the points B, D, G 
and K, back into the bases and through them draw the elements. Their intersection at 3, 4, 5 
and 6 will be the four points determined by the plane VW. In like manner the points 7 and 
8 may be found by using a plane tangent to the small cylinder. The points in the top view 
are obtained by projecting them from the front view on to the plane in which they lie. Thus 
the point 7 was found by means of the cutting plane XY, and its top view, 7", must lie on the 
line XY which is the horizontal trace of the plane. In the case illustrated by Fig. 162, it would 
be well to use three cutting planes hetween VW and XY. This will determine a sufficient 
number of points to sketch the curve accurately. 


THE USE OF AUXILIARY PLANES 99 














100 


THE INTERSECTION OF SURFACES 


125. The intersection of an oblique and a vertical cylinder. Either of the two pre- 
ceding methods may be employed in the solution of this problem. The side view not being 
available for spacing the elements, the following method may be pursued: Draw any element 





Fig. 165. 


of the small cylinder, as AB, and let XY be a circle of the 
cylinder made by a cutting plane perpendicular to the axis. 
If this circle be revolved about XY parallel to the vertical 
plane, the point C, which is a point of both the element and 
the circle, will be revolved to C’, and CC’ will be the dis- 
tance between the center line and the element in the top 
view, the latter piercing an element of the large cylinder at 
A¥#, AY. Since there is a second element, ED, of the small 
cylinder lying in the same vertical plane as AB, it will inter- 
sect the same element of the large cylinder at the point DY. 
Thus find any number of points. It will be convenient to 
have the elements of the small cylinder spaced equally, and 
this may be done by spacing them on the revolved position 
of the circle XY, and through these points drawing the 
elements. In developing the small cylinder it will be found 
necessary to assume some line of the surface which -on being 
developed will be a straight line, for in this problem the 


development of the ends of the cylinder will be curves. Such a line wili lie in a plane 
perpendicular to the axis, and the section of the cylinder made by this plane is called a 
RIGHT SEcTION. The line XY fulfills this condition and may be used as a base line for 


THE INTERSECTION OF PRISMS 101 


measuring the length of the elements in the development, and on its revolved position, 
XC'Y, the distances between the elements may be measured. 


126. The intersection of prisms. In the intersection of two prisms, or two pyramids, or 
a prism and a pyramid, the line of penetration will be a broken line instead of a curve. As 
auxiliary cutting planes cannot be used to advantage, it is necessary to find the points of 
intersection of each edge of the one with a face of the other. 
To avoid confusion the points should be obtained consecutively. 

Fig. 166 illustrates two intersecting hexagonal prisms. 
Having drawn the H and V projections, determine the points 
in the following manner: The edge 1 will intersect the upper 
face of the vertical prism at the point K. K™®™ is obtained 
directly from the V projection, KY. Next obtain the point 
of intersection of the upper edge of face AB with face 1 2. 
The plane of the upper face of the vertical prism will cut the 
face 1 2 in the line KS, and, since the upper edge of face AB 
lies in this plane, its intersection with KS, at L, will be the 
desired point. The next point of intersection, M, of the edge 
2 with face AB is apparent from the top view, M® being the 
H projection of this point. The point N, which is the inter- 
section of edge 3 with face AB, is similarly obtained. The 
next point to be determined is O, the intersection of edge A 
with the face 3 4. This completes one-half the intersection. 





CHAPTER IX 
SPIRALS, HELICES, SCREW-THREADS AND BOLT-HEADS 


127. The Spiral is a curve generated by a point moving in a plane about a center from 
which its distance is continually increasing. 

Imagine a right line, AB, Fig. 167, free to revolve in the plane of the paper about one of 
its extremities, A, as an axis. Also, conceive a point free to move on this line. Three classes 
of lines may now be derived in the following manner: If the line be stationary and the 
point moves, a straight line will be generated; If the point be stationary and the line revolves, 
a circle will be generated; If both point and line move, a spiral will be generated. By vary- 
ing the relative motion of point and line the character of the curve will be changed and the 
various classes of spirals described. 


128. The Equable Spiral. Fig. 167. If the motion of the line and the point be uniform 
_the equable spiral will be generated. This is also called the SPIRAL OF ARCHIMEDES. The line 
‘AB is called the RApIus VEcTOR, and the radial distance traversed by the point during one 
revolution of the radius vector is called the PircH. ‘T'welve successive positions of the radius 
vector are shown by AC, AD, AE, etc. The distance of the point from the center being 
increased by one-twelfth of the pitch, AC, for each one-twelfth of a revolution of the radius 
vector. In practice, determine at least twenty-four points, and lightly sketch the curve free- 
hand. ‘This is the curve used in the cam for uniform motion. Instruction for the inking of 
spirals is given in Art. 15, page 12. | 
102 


THE LOGARITHMIC SPIRAL 103 


129. The Equiangular or Logarithmic Spiral. The 
construction of this curve is based on the principle 
that any radius vector, as AD, Fig. 168, which bisects 
the angle between two other radii, as AC and AB, is 
a mean proportional between them; that is, AD*= 
AC x AB. This spiral is called equiangular because 
the angle between any radius vector and the tangent 
to the curve at its extremity is constant. 

If B and C are points in the spiral and the ratio 
of AC to AB be given, the intermediate point D may 
be obtained by describing a semicircle on BC as a 
diameter and erecting a perpendicular at A. Its 
intersection with the semicircle at D will determine 





the required point, and AD will be a mean propor- 
tional between AC and AB, Other points on the 
curve lying on the diameters BC and DF may be 
obtained by intersecting these diameters with the per- 
pendiculars CL, LM, MN, ete. Again, having points 
C and D, bisect DAC and determine a mean propor- 
tional between AC and AD. This may be done by ; 
laying off AF equal to AC and determining the mean ss Fig. 168. 
proportional AK as before. ‘Then lay off AE equal to : 
AK. Other points on the diameters ER and ST may be obtained by perpendiculars as indicated. 





SPIRALS, HELICES, SCREW-THREADS AND BOLT-HEADS 





Fig. 171. 


130. Involutes. This class of spirals may be generated by un- 
winding a perfectly flexible but inextensible cord from a polygon of 
any number of sides, the names of the involutes being derived from 
the polygons which determine their form. ‘The curves consist of 
circular arcs having the vertices of the polygons for centers, and 
radii increasing by an amount equal to the length of the sides. 
Fig. 169 is the involute of a square, drawn by describing the 
quadrant 4-5 from center 1 with radius 1-4; then the quadrant 5-6 
from center 2 with radius 5-2, and so on until the desired length of 
curve shall have been drawn. If two opposite sides of the square 
become infinitely small, the result will be a right line, the involute 
of which is shown in Fig. 170. Again, if the number of sides of the 
polygon become infinite, we shall obtain the involute of a circle, as 
in Fig. 171. The accuracy of this curve will depend on the number 
of points in the circle; in this case there are twenty-four. 


131. The Helix. If the line on which the generating point is 
supposed to move be made to revolve about an axis with which it 
makes an angle of less than 90°, thus generating a cylinder or cone, 
a class of curves known as helices will be described. If the line 
AB, Fig. 172, be parallel to the axis about which it revolves, and 
the generating point moves on this line, three classes of lines may 
be described, as in the case of spirals. <A circle will be generated 


THE HELIX 


when the point is stationary and the line revolves; a right line 
will be generated when the line is stationary and the point moves on 
the line; and since the circle and right line do not le in the same 
plane, the result will be a helix when these motions take place simul- 
taneously. ‘The distance traversed by the generating point on the 
line AB during one revolution is called the PircH. The motion of 
line and point being in general uniform, the curve is described as 
follows: Having assumed any desired number of equidistant posi- 
tions of the generating element AB, as 1, 2, 3, 4, etc., the pitch 
should be divided into the same number of parts, as shown by the 
horizontal lines drawn through 1’, 2’, 3’, 4’, ete. When the element 
shall have moved through one of its divisions to the position 1, which 
in this case is one-twelfth of a revolution, the generating point will 
have moved on the generating line through one-twelfth of the pitch, 
and the point K will be determined. Also, when the element has 
made one-quarter of a revolution, the point will have traversed one- 
quarter of the pitch and be at C. As the rate of curvature is most 
rapid at the points of tangency, A, D and B, it is desirable to obtain 
a greater number of points by subdivision, as shown in the figure. In 
Fig. 172 the helix is right-handed, and single, but if a double helix is 
required, draw a parallel curve beginning at the point 6’, opposite D. 

A conical helix is generated by the motion of a point on an ele- 








~ 


—~ > 


~ 





~ 





~ 





~ 


Qa &o N 


~ 











Fig. 172. 


ment of acone. ‘The pitch is measured parallel to the axis, as in the preceding case. 





106 SPIRALS, HELICES, SCREW-THREADS AND BOLT-HEADS 


132. Screw-Threads. In order to make the drawing of a screw-thread, it is necessary to 
know the diameter, pitch and section of the thread. 

If the thread to be drawn has a V section, as in Fig. 174, and the diameter and pitch are 
given, begin by drawing the section of the screw as shown in dotted lines. Next, construct a 
templet as follows: Draw on a rather light piece of cardboard two helices of the same pitch 
as the required thread, one being of a diameter 
equal to the outside of the thread and the other 
equal to the diameter at the root of the thread. 
These may be drawn separately, or together, as 
in Fig. 178, and are to be carefully cut out so 
as to be used as a pattern in pencilling the curves. 
Having drawn the helices, mark the points of 
tangency B, G, and A, F, as well as the centers 
Fig. 173, L and K, in the manner indicated. Also repeat 
| these marks on the opposite side after cutting 

out. This will enable the templet to be used 
either side up, and be readily set to the drawing. Observe that the curve is not to be cut off 
abruptly at its termination, but continued a little beyond, so that in tracing the outline the 
pencil-point may not injure the extreme point of the templet curve. This may now be used 
for the drawing of all the helices on this screw, as from A to F, B to G, C to H, etc., Fig. 174. 

If the pitch is small in proportion to the diameter, the drawing of the screw may now be 
considered finished; but the contour line does not coincide with that of the section of the 
thread, and in order to illustrate correctly the projection of a V thread we must consider the 


























L— | K 


SCREW-THREADS 


character of the surface and apply a correction to the 
drawing. The surface which is being drawn is a 
helicoid, and is generated by the motion of a line, 
AB, which is made to revolve about the axis of the 
screw and at the same time move in the direction of 
the axis. ‘This will generate the upper half of the 
surface of the screw. Every point of the line will 
describe a helix, the diameters differing, but the pitch 
remaining constant. The helices generated by the 
extremities of the line have already been drawn and 
the curves 1 2 3 and 4 5 6, described by two other 
points, 1 and 4, are shown by the fine dotted lines. 
The curved line M5 2N, drawn tangent to these 


helices, will be the visible outline of the surface . 


instead of the dotted line which is concealed. As 
the labor of describing these helices would be great, 
it is customary to draw the outlines of the screw as 
follows: Having described the helices AF, BG, 
etc., reverse the templet and draw a small portion of 
the continuation of the helix on the opposite side of 
the screw, as at BP and CO. Then, draw the line 
MN tangent to the two helices, and in the other 
direction the tangent OP, a part of which is invisible. 


107 








108 SPIRALS, HELICES, SCREW-THREADS AND BOLT-HEADS 


Be sD 


Fig. 176. 





133. Conventional V Threads. In order to facilitate the 
drawing of screw-threads it is customary to omit the drawing 
of the helix, substituting therefor a right line, as in Fig. 175. 
In most cases, however, even this would involve too much labor 
as well as complication on the drawing, and the V’s are likewise 
omitted, the representation shown in Fig. 177 being adopted. 
When this is done, no care is taken to make the screw of the 
required pitch, and the spaces between the fine lines which 
represent the outer helices are estimated by the eye, as in 
section lining. But it is imperative for the proper representa- 
tion of a single thread, that the point C be over the middle of 
the space BD. Having drawn the line AB, make the space 
AC double that of EB and draw parallels. Afterwards, draw 
the heavy lines to represent the root of the thread. As it is 
difficult to make these of equal length without the aid of special 
lines, the method illustrated in Fig. 178 is frequently used. 


134. The Double Thread. As the use and character of the 
double thread are generally misunderstood, Figs. 175 and 176 
have been drawn to explain this problem more clearly. Fig. 
175 illustrates a screw, the diameter and pitch of which are 
supposed to have been given; but, as the pitch is excessive for 


a screw of this diameter, the diameter at the root of the thread is small, and the screw propor- 


U.S. STANDARD V THREADS 109 


tionally weak. The only way to strengthen the thread at this 
point without changing the angle of the V’s is by partly filing 
the V’s at the root, as shown by the dotted line in Fig. 175 
and the left-hand portion of the complete screw in Fig. 176, 
thus increasing the diameter at the root. While overcoming 
one weakness we have introduced a second by lessening the 
section of the thread, so that with a nut of a given length the 
tendency of the thread to be stripped from the body is doubled. 
This last difficulty may be overcome by supposing an inter- 
mediate thread wound between the present threads, as shown 
by the right-hand portion of Fig. 176, which is a representa- 
tion of a double thread having the same diameter and pitch 
as the single thread of Fig. 175, but of increased strength. It 
must be noted that the threads indicated by AB and CD, of 
Fig. 176, are entirely independent of each other and that the 
point C of one is diametrically opposite a point B in the par- 
allel thread. This must be carefully observed-in the practical 
representation of a double thread, as shown in Fig. 179. Fig. 
180 represents a left-hand single thread. 


_ 135. U.S. Standard V Threads. ‘The form of thread com- 
monly used is that of the U.S. Standard, also known as the 
Franklin Institute Standard and illustrated by Fig. 181. 





RIGHT HAND SINGLE THREADS 


Fig. (77. Fig. 178. 


ANY 0H 


DOUBLE THREAD LEFT HAND THREAD 


Fig. 179. Fig. 180. 





P= 0.24VD + 0.625 — 0.175. 
S = 0.65 P. 


110 SPIRALS, HELICES, SCREW-THREADS AND BOLT-HEADS 





Although the pitch in a single-threaded screw is the: 


distance between consecutive threads,. the term is often 
applied to the number of threads per inch. Thus a screw 
having eight threads to an inch is frequently spoken of as 
8 pitch. This is obviously wrong, but leads to no confu- 
sion, since the pitch and the number of threads per inch are 
reciprocals of each other. The flattening of the thread, as 
indicated in Fig. 181, is for the purpose of preventing injury 
by the bruising of the otherwise sharp V. 


136. Square Threads. Fig. 184 represents a thread of 
square section. ‘The size of this square is equal to one-half 
the pitch in a single thread, and one-quarter of the pitch 
in a double thread. The construction is similar to that 
described for the V thread. The outline of the section is 
drawn first, and a templet is prepared for the inner and outer 
helices. The threads are drawn as in Fig. 184 for a single 
right-hand thread, and as in Fig. 185 for a nut in section. 
Figs. 186 and 187 illustrate a double square thread and nut. 

The conventional representation of the square thread is 
similar to that of the V thread, the right line being substituted 


for the helix. Fig. 182 is a correct drawing of the conventional square thread, and Fig. 183 
a modification more commonly used. 


< 








Ek 
= 
(ea 


Corea 
Pars 


<q 

Tk 

= 

she 
: eet 






\NS )\ 





137. Sphere and Cutting Planes. If a sphere be cut 
by six planes equidistant from and parallel to a vertical 
axis, a representation will be obtained similar to that of a 
~ hexagonal bolt-head or nut. Since all planes intersect a 
sphere in circles, these intersections will appear as circles, 
ellipses or right lines. Such a representation is shown by 
Vig. 188, VW and XY being two of the cutting planes. 
The plane V W intersects the sphere in a circle the diame- 
ter of which, is E#F#, and shown on the front plane by 
EVYAYBYFY, The plane XY intersects the sphere in a 
circle of equal diameter, being 
at the same distance from the 
axis; but,as this planeis inclined 
to the V plane of projection, the 
circle will appear as an ellipse 
having a major axis, GY HY, equal 
to the diameter of the circle, and 
the minor axis, KYLY, equal to 
the foreshortened diameter pro-. 
jected from the top view. ‘The 
planes VW and XY intersect 
in the line BYOY, thus cutting 
off a portion of the circle and ellipse. Similarly the remaining curve of intersection, DYAY, 














SPHERE AND CUTTING PLANES 113 


may be found and the side view likewise determined. Points 
in the curve may also be obtained by revolving the plane 
XY parallel to H, thus obtaining the height of any point, 
as N, which may then be projected into the other views. 

Since the upper half of the two vertical projections of 
Fig. 188 may be regarded as a true representation of a 
hexagonal bolt-head, save as to proportion, it is important 
to consider the salient points of these projections. 

First: Three faces of the head are seen when it is 
shown “across corners,” as in the front view, and one of these 
is double the width of the other two. 

Second: The circular arc AYMYBY is concentric with 
the circle of the sphere, and the points AY and BY are deter- 
mined from the height of CY, the intersection of the two 
planes or faces of the head and the great circle of the sphere. 

Third: The major axes of the ellipses and the diam- 
eters of the circles made by the cutting planes are equal, 
hence points GY and MY are at the same height. 

Fourth: Two equal faces are seen when the head is 
shown “across flats,” as in the side view. 

Fifth: The points MP and GF are of equal height, as 
in, the front view, and B® must be obtained by projection 


nS ea 
from the front view or in the same manner as CY. +! 


Fig. 189 is a sphere cut by four planes‘and similar to a square-headed bolt or nut. 





114 SPIRALS, HELICES, SCREW-THREADS AND BOLT-HEADS 


kK LENGTH 
1 


Lane D=DIAMETER OF BOLT a eee 
ie - | 
Ty 
My 
Fig. 191. | ee) 





I< LENGTH 





138. U.S. Standard Hexagonal Bolt-head 
and Nut. ‘Two types of head and nut are 
illustrated, the rounded or spherical, Fig. 
190, and the chamfered or conical, Fig. 191. 
Three dimensions are fixed by this standard: 
First, the distance across flats or short diame- 
ter, commonly indicated by H, and equal to 
one and one-half times the diameter of the 
bolt plus one-eighth of an inch: Second, the 
thickness of the head, which is equal to one- 


half the short diameter, or = Third, the 


thickness of the nut, which is equal to the 
diameter of the bolt. 

Suppose it is required to draw a rounded 
head “ across corners,” as shown in Fig. 192. 
Having drawn the center line, underside of 
head and diameter of bolt, as indicated by 
lines 1, 2, 8 and 4, figure the short diameter 
of the hexagon or distance “across flats” 
according to the proportion given. Lay off 
EF equal to one-half this amount and draw 
the perpendicular FG and the 30° line EG, 


BOLT-HEAD AND NUT EES 


then will the triangle EFG represent one-twelfth of the top view 
of the head, and EG will be equal to one-half the long diameter 
required. Lay off this distance on either side of E, thus deter- 
mining lines 8 and 9. Draw 10 and 11, remembering that these 
lines equally divide the spaces between 1 and 8, and 1 and 9, which 
spaces also equal twice FG. Next determine the thickness of head, 
and with a radius * equal to twice the diameter of the bolt, describe 
are 12, which determines points D, C, A and B. From the same 
center describe arc 14. Arcs 15 and 16 should be drawn as cir- 
cular arcs, their height being determined from 14. 

If it is required to represent a bolt-head “across flats,” as in 
Fig. 193, proceed as before, determining the short diameter and 
drawing 5and 6. Next lay off the thickness of head and describe 
arc 7; this will determine E and the height of arcs 12 and 13. 
- Although these arcs are elliptical in theory, they should always 
be described as circular. ‘To determine the point A, find the long 
diameter and obtain line 10; its intersection with 7 will be at D, 
equal in height to A, Band C. Finally draw arcs 12 and 13. 

Fig. 194 illustrates a rounded nut “across corners,” the order 
for the drawing of the lines being indicated by the figures. The 
thickness of a nut is equal to the diameter of the bolt. As the nut 
is pierced by a hole, the top will appear flat and the are 13 must 
be drawn from K, but with radius equal to 2D. 





194. 


* There is no standard for this radius, but 2D is recommended as being a convenient radius for draftsmen. 


116 SPIRALS, HELICES, SCREW-THREADS AND BOLT-HEADS 


139. Chamfered Head and Nut. If we substi- 
tute a cone for the sphere in Fig. 188, page 112, 
cutting it by the six vertical planes parallel to the 
axis, and then pass a seventh plane perpendicular 
to the axis and tangent to the curves of intersection, 
a representation will be obtained similar to that of 
Figs. 195 and 196. ‘This is called a chamfered head, | 
and the curves of intersection are hyperbolas; but, 
since these curves approximate circular ares, the fol- 
lowing concise method may be employed. If it is 
required to draw a chamfered head or nut ‘across 
corners,” construct a hexagonal prism of dimensions 
required for a standard bolt. From the center line, 
with a radius equal to diameter of bolt, describe the 
arc AB tangent to top of head, and from the same 
center draw arcs DE and CF, points D and C being 
determined by A and B. Finally draw ares DA 
and BC tangent to EF. The method for drawing 
the chamfered head “across flats” is apparent from 
Fig. 196. A bolt with chamfered head and nut is 
illustrated by Fig. 191. 

Fig. | 196. For the further consideration of this subject the 
student is referred to the chapter on Bolts and Screws 
in “ Machine Drawing ” of this series. 








CHAPTER X 
PROBLEMS 


_ 140. General instructions for performing the problems. Size of paper 11” x15!. Art. 
26, page 20. Space within the margin lines 10’ x14./’ Art. 28, page 21. The space 
required for performing the problems is given whenever the graphic statement is not made. 
All dimensions are from the margin lnes, and in the graphic statements these are shown. 

Read the articles noted before beginning to solve a problem. Note every detail of the 
graphic statement and observe the method required for the solution of the problem. It is 
not intended that the order of the problems shall necessarily be followed or that all of 
the problems shall be performed, but that a judicious selection be made to meet the vary- 
ing needs of students. Additional problems of a more practical character may be found 
among the miscellaneous problems of Art. 150, page 149. 


141. The use of instruments. Examples for practice. The only benefit to be derived 
from the making of these drawings will be in gaining a knowledge of the use of instruments ; 
but if the directions are not observed and the prescribed methods carefully followed, much 
of the value of this study will be lost. These problems occupy a space of 4!’ square. Six 
may be drawn on each plate as indicated by the figures. Read Chapters I and II, together 
with such additional articles as may be prescribed in connection with the problems. 

117 


118 

















PROBLEMS 


Pros. 1. By means of a triangle and T 
square draw AB and CD, and divide them into 
quarter-inches by the scale. Art. 18, page 8. 
Through these points draw parallel lines by the 
aid of a straight-edge or triangle, but do not 
use the T square. ‘Their parallelism may be 
tested by shding one triangle on another. 

Pros. 2. Having drawn the square, divide 
the left-hand edge into 10 equal divisions by 
dividers or by Art. 40, and draw horizontal lines 
with the T square through these points. In 
inking these lines observe the variety and grade 
indicated by Fig. 30, Art. 29, page 22. 

Pros. 8. Divide EF and FG into half- 
inches and divide the surface into squares. In 
drawing the intermediate lines, judge the spacing 
by the eye and draw the upper horizontal lines 
in the first, third, fifth and seventh squares ; 
then the second horizontal lines in like manner, 
and so proceed until all the horizontal lines are 
drawn. Similarly draw the vertical lines. See ~ 
that the lines begin and end exactly on the 
required points or lines. 


THE USE OF INSTRUMENTS 119 


Pros. 4. Use the T square and 45° triangle to divide the square as indicated, the size of 
the small squares being 3/’.. This is an excellent test of precision in measurement and lining. 

Pros. 5. The lower edge of this square is to be divided into half-inches and con- 
verging lines drawn to the upper corners. Care must be observed in the inking of this figure 
to allow each line to dry before inking the next one. 

Pros. 6. This is a practice similar to that of Prob. 5. 

Pros. 7. Divide the square as indicated by the dotted lines, drawing these as full lines 
in pencil, but do not ink them. These divisions will facilitate the drawing of the figures by 
making the use of a scale unnecessary. ‘The intermediate or section lines must not be 
drawn with a pencil, but are to be drawn only in ink, the spacing being done by the eye. 
Art. 32, page 27. Ink the fine lines first, then the shade lines, and finally the section lines. 
Art. 30, page 23. 

Pros. 8. Observe the instructions for Prob. 7. 

Pros. 9. As the divisions of the square are not easily obtained by the scale, use the 
dividers as in Art. 18, page 14. Observe the instructions for Prob. 7. 

Pros. 10. The lines of this figure are to be drawn by means of the 45° triangle on the 
T square. In pencilling, it will be found to be more convenient to draw one set of lines 
continuous, as the invisible portions of the bands are not readily determined at first. This 
figure should be shaded. 

Pros. 11. The sides of this square are to be divided into six parts, which may be done 
by the use of dividers. Art. 18, page 14. This figure may be shaded and section lined as 
in Prob. 7, observing the directions given for that problem. 

Pros. 12. Observe the directions for Prob. 11. 


120 PROBLEMS 


Pros. 18. Divide the horizontal diameter 
into quarter-inches and draw eight concentric 
circles through these points. Arts. 16 and 17, 
page 12. Use great care to avoid enlarging the 
center. Ink with fine lines. 

Pros. 14. Draw eight concentric circles 
in full pencilled lines. Ink by a fine dotted 
line. Allow sufficient time for the drying of 
the ink in each circle before drawing the next. 

Pros. 15. Divide the horizontal lines 
into half-inches and draw a series of circles 
through these divisions tangent to the outer 
circle. In inking, begin with the smallest 
circles and allow time for the ink on each to 
dry before inking the next circle. Use the 

a bow instruments for small circles. Arts. 19 

eRe and 20, page 16. 

BERG) EEROB. itine Divide the horizontal lines 
Yj) OG into half-inches and draw semicircles in the 
CAG following order: AB, HK; AC, GK; AD, 
yy FK; etc. See that the circular arcs make con- 
Yyey, —MVi, ti tinuous lines and are tangent to the outer circle. 
oo Ink the lines in the same order. 






































RW 


AN 





Nal 


Sy 





SS 





9 


Ba! 


THE USE OF INSTRUMENTS . 121 


Pros. 17. Similar to Prob. 16, save that circles are required instead of semicircles. 

Pros. 18. Draw a circle of 2! diameter and divide it into twelve equal parts. From 
these divisions describe arcs, each passing through the center and terminating in the pre- 
ceding arc. ) 

Pros. 19. Make a 4!’ square and divide it as follows: With a 45° triangle draw lines 
AC and BD through the center and construct three squares. Set the bow-pencil to a radius of 
3" and describe a circular are in the corner of each square just touching, but not intersecting, 
the sides of the square. Ink the circular ares first. | 

Pros. 20. Divide a 4’’ diameter into half-inches and describe circles as indicated. Do 
not shade the lines in pencilling. In inking this figure, suppose it to represent a series of rings 
of $’’ width and shade by the method of Art. 30, page 25. 

Pros. 21. This is a practice in the shading of small circular ares with the bow-pen. 


Subdivide a 4’ square as indicated and describe circles of 3/’ diameter, shading them as in 


Art. 30, page 26. | 

Pros. 22. Draw the figure prescribed for Prob. 19 and, in inking, shade as in Fig. 88, 
page 26, supposing the figures to represent a series of hollow squares. 

Pros. 253. This is a practice in shading and section lining. Construct according to 
dimensions and carefully observe the directions for shading. Art. 30, page 23. The section 
lining must be done only in ink. 

Pros. 24. This is a practice in the joining of shaded arcs, the smaller of which should be 
drawn with the bow-pen, and the larger with the compasses. The section lining must be done 
only in ink. 


122 PROBLEMS 


142. Geometrical problems. A 4!’ square is sufficient space for each problem. ‘These 
squares may be arranged as illustrated on page 118, allowing six to each plate. All of the 
work is to be performed with a 4H lead pencil. The greatest possible precision must be used. 
Construction lines are to be made very fine; given and required lines being made stronger by 
pencilling a second time. When two methods are given, the problem should be constructed 
by the draftsman’s method and tested by the geometrical. method. It is desirable thoroughly | 
to master the propositions relating to the drawing of perpendiculars and then perform the 
problems involving these principles. Next, study those relating to angles and similarly per- 
form the dependent problems. ‘Thus continue the subject according to the divisions indicated. 
Place the number of the problem in the right hand lower corner of the square containing it. 

Do not ink the problems, as it impairs the accuracy of the work. If practice in inking is 
desirable, the figures of the preceding article are better adapted to this exercise. 

1. Bisect a line 3” long. Art. 39, page 36. 

2. Bisect an arc of 23” radius and 23” chord. Art. 39, page 36. 

3. Draw a line 27’’ long and erect a perpendicular 13’ from one end. Do not use a T © 
square. Art. 41, page 36. 

4. Draw a line 2,°,/’ long and erect perpendiculars at the extremities. Use both methods. 
Art. 41, page 36. 

5. From a point nearly over the center of a line 5} long draw a perpendicular to the 
line. Art. 41, page 87. 

6. From a point nearly over the extremity of a line 2,%'’ long draw a perpendicular to 
the line. Do not use a T square. Art. 41, page 37. 


GEOMETRICAL PROBLEMS | 123 


T. Draw two intersecting lines making any angle. Construct a similar angle and bisect 
it. Arts. 48, 44, page 38. 

8. From one extremity of a line 33/’ long draw a second line making an angle of 45° with 
the first. Similarly construct an angle of 30° at the other end. . Arts. 42, 45, 46, page 38. 

9. From one extremity of a line 3}/’ long draw a line making an angle of 223° with it. 
Similarly construct an angle of 15° at the other end. Arts. 42, 43, 45, 46, page 38. 

10. From one extremity of a line 32’! long draw a second line making an angle of 60° 
with the first. Similarly construct an angle of 75° at the other end. Arts. 42, 45, 46, page 38. 

11. Describe a circle 3}’’ in diameter and divide it into angles of 15° by means of triangles 
and aT square. Art. 42, page 38. 

12. Construct an equilateral triangle having a base of 

13. Construct an isosceles triangle having a base of 
48, page 39. 
. 14. Construct an isosceles triangle niet a base of 13/’ and the equal angles 75°. Art. 48, 

page 39. 

15. Construct an isosceles triangle having a base of 3%’, the angle at the vertex being 
150°. Art. 48, page 39. 

16. Construct a scalene triangle having sides of 2}'’, 23’ and 313". Art. 49, page 39. 

17. Construct a right-angle triangle having a base of 2a ‘and one angle of 30°. 

18. From a point on a circle 27’ in diameter draw a finpene Arts. 50, 51, page 40. 

19. Draw a tangent to the middle point of the are of a circle of 24” radius having a chord 
of 24’. Do not use the center of circle. Art. 51, case 2, page 40. 


215", Art. 47, page 389. 
24/’ and the equal sides 33!", Art. 


124 | PROBLEMS 


20. Draw a tangent to a circle 23’’ in diameter from a point 24’ from center of the 
circle. Art. 51, case 8, page 40. | 

21. With a 33” radius describe an arc of 45°, and from one of its extremities draw a_ 
tangent equal to the length of the arc. Art 53, page 41. From the point of tangency lay off 
24” on the tangent and obtain an are of equal length. Art. 52, page 41. 

22. Draw two lines making an angle of 45° with each other, and a circle tangent to these 
lines at a point 13’’ from vertex of the angle. Art. 54, page 42. 

23. Draw two lines making an angle of 30° with each other, and two circles tangent to 
each other and these lines. ‘The diameter of the smaller circle is ?/’. Art. 55, page 42. 

24. Draw a tangent to two circles having diameters of 13/’ and #/’, and their centers 2! 
apart. Art. 56, page 42. 

25. Draw a circle having a diameter of 14!’ tangent to two circles having diameters of 
13" and 12” with centers 13” apart. Art. 57, page 43. 

26. Prescribe three points and draw a circle through them. Art. 58, page 43. 

27. Circumscribe a circle about a scalene triangle having sides 13/’, 23/’ and 23/’. Art. 
59, page 45. 

28. Inscribe a circle within an isosceles triangle having a base of 34/’ and equal sides of 
38/’, Art. 60, page 44. 

29. Draw a right-angle triangle having a side 23/’ long and one of the oblique angles 30°. 
Circumscribe a circle about this triangle. 

30. Within a circle 32” in diameter inscribe an equilateral triangle. Art. 61, page 44. 

381. Within a circle 33/’ in diameter inscribe a square. Art. 62, page 44. 


GEOMETRICAL PROBLEMS 125 


32. Within a circle 33’ in diameter inscribe a pentagon. Art. 63, page 44. 

33. Within a circle 33” in diameter inscribe a hexagon. Art. 64, page 45. 

a4. About a circle 2%” in diameter circumscribe a hexagon. Art. 65, page 45. 

80. Draw a hexagon having its long diameter 33'’. Art. 66, page 46. 

36. Draw a hexagon having its short diameter 3/’.. Art. 67, page 46. 

37. Draw a hexagon having one side 18/’. Art. 68, page 46. 

38. Draw a hexagon having one side 13’’ long and at an angle of 45° with the horizontal. 

39. Draw a hexagon having its short diameter 23/’ and one side horizontal. Art. 67, 
page 46. 

40. Within a circle 34/’ in diameter inscribe an octagon. Art. 69, page 47. 

41. Circumscribe an octagon about a circle 3”’ in diameter. Art. 70, page 47. 

42. Within an equilateral triangle having sides 33/7 draw 3 equal circles touching each 
other and one side of the triangle. Art. 72, page 48. 

43. Within an equilateral triangle having sides 33!’ draw 8 equal circles touching each 
other and two sides of the triangle. Art. 73, page 48. 

44, Within an equilateral triangle having sides 33?!’ draw 6 equal circles which shall be 
tangent to each other and the sides of the triangle. Art. 74, page 48. 

45. Within a circle 33’’ in diameter draw 3 equal circles tangent to each other and the 
given circle. Art. 75, page 48. 

46. Within a circle 3?’ in diameter inscribe 5 equal circles tangent to each other and the 
given circle. Art. 76, page 49. 

47. About a circle 14'’ in diameter circumscribe 5 equal circles tangent to each other and 
the given circle. Art. 77, page 49. 


126 PROBLEMS 


143. Conic Section Problems. These problems require a space of 5!’ x 7’’, or four problems 
to each plate. Use very fine full lines for all construction, but do not ink them. ‘The required 
curves should be inked with care, observing the instructions in Art. 15, page 10. 

Study Arts. 78 to 84 inclusive, before performing the following problems on the ellipse. 

1. Draw an ellipse having the minor axis 4" and the distance between the foci 43. Use 
the First Method. Art. 80, page 52. Draw the conjugate diameters, one of which makes an 
angle of 15° with the major axis. Art. 79, page 52. 

2. Draw one-half of an ellipse having the major axis 63’ and the distance between the 
foci 43". Use the Fourth Method. Art. 83, page 565. 

3. Draw an ellipse having the major axis 63/’ and the minor axis 3/’. Use the Second 
Method. Art. 81, page 54. Draw a tangent to the curve at a point 3! from the center. 
Page 53. | 
4. Draw an ellipse having the major axis 5} and its minor axis 43’. Use the Fifth 
Method. Art. 84, page 56. ‘Test one-quarter of the curve by the Fourth Method. Art. 83, 
page 50. 

5. Draw an ellipse having the major axis 53’ and the distance between foci 33”. Use 
the First Method. Art. 80, page 52. Draw two conjugate diameters, one of which makes an 
angle of 75° with the major axis. Art. 79, page 52. 

6. Draw an ellipse having the minor axis 33!’ and the distance between the foci 5M, Use © 
the Second Method. Art. 81, page 54. Draw a tangent to the curve at a point 2?” distant 
from the minor axis. Page 58. 

7. Draw one-half of an ellipse having its major axis 63”’ and its minor axis 44/’, Use the 
Fourth Method. Art. 83, page 55. 


CONIC SECTIONS 127 


8. Draw an ellipse having the major axis 4%/’ and the minor axis 23’. Use the Third 
Method. Art. 82, page 54. Test four points by the First Method. Art. 80, page 52. 

9. Draw an ellipse having the major axis 6’’, and the minor axis 4’. Use the First 
Method. Art. 80, page 52. Draw a tangent to the curve at a point 23’ from one extremity 
of the minor axis. Page 53. 

10. Draw an ellipse having its minor axis 4’’, and the distance between the foci 3’. Use 
the Second Method. Art. 81, page 54. Draw two conjugate diameters, one of which makes 
an angle of 30° with the major axis. Art. 79, page 52. 

11. Draw a segment of an ellipse having an axis of 4/’ and a 53” chord which intersects 
the axis at 1}/’ from its extremity. Use the Fourth Method. Art. 83, Case 2, page 55. 

12. —_ one-half of an ellipse having a major axis of 4'’ and distance between the foci 
2". Use the Third Method. Art. 82, page 54. Draw the meena half of the ellipse by the 
Fifth Method. Art. 84, page 56. Test the ellipse by the Second Method. Art. 81, page 54. 


Read Arts. 85 to 88 inclusive, before performing the following problems on the parabola. 
Draw the axes horizontal. Ink the curves, but not the construction lines. 

13. Draw a parabola having the focus 3?’ from the directrix. Draw a tangent to any 
point of the curve. Art. 86, page 57. 

14. Draw a parabola having given the abscissa 6’’ and double ordinate 44’. Art. 87, 
page 58. | 

15. Draw a parabola having the focus 1’ from the directrix. Draw a tangent to the 
curve at a point 13’ from the focus. Art. 86, page 57. 

16. Draw a parabola having given the abscissa 54’! and double ordinate 4/’.. Art. 87, 
page 58. se 





128 PROBLEMS 


Read Arts. 89 to 93 inclusive before eae is the following problems on the hyperbola. 
Draw the transverse axes horizontal. 

17. Draw an hyperbola having its transverse axis 14/’ and distance between the foci 24 = 
Draw a tangent to the curve at a point 13’ from the vertex. Art. 90, page 59. 

18. Draw an hyperbola having its transverse axis 13/’, a double ordinate 43/7 and its dis- 
tance from the vertex 1f/’.. Art. 91, page 59. 

19. Draw an Hemerbole having its transverse axis 1}/’ and distance between the foci 
24’. Draw a tangent to the curve from a point 14’ from the vertex. Art. 90, page 59. 

20. Draw an hyperbola having the transverse axis 13/’, a double ordinate 43’ and its dis- 
tance from the vertex 24’. Art. 91, page 59. 

144. Orthographic Projection Problems. Most of the problems are designed to occupy a 
space of 5!’ x 7/’, in which case there will be four problems to each plate and they should be 
separated by a fine inked line. All construction lines and lines of the object should be drawn 
very fine in pencil and no line that has been useful in the construction of the drawing should 
be erased. Invisible lines of the object are better dotted in pencil to avoid mistakes in inking. 
Draw no dimension lines. Study Arts. 94, 95 and 96. 

Begin the drawing with that view concerning which the most information is given. It is 
not necessary to complete one view before beginning a second, and frequently it is desirable to 
proceed with the three views at one time. Carefully estimate the space to be occupied by each 
view when a graphic statement of the location is not given. 

Only lines of the object are to be inked, the visible lines being in full and the invisible in 
dotted lines. Shade lines are to be used only in those problems indicated, and never shown 
in pencil. Observe the instructions for inking. Art. 33, page 28. 


ORTHOGRAPHIC 


Pros. 1. Locate the ground line and 
traces of profile plane. Draw top and front 
views, and from these obtain the side view. 
Draw projection linesin pencil only. It is an ex- 
cellent practice to number the extremity of each 
line of the completed projection as in Fig. 107, 
page 65, and thus acquire familiarity with the 
different views of each surface, line and point. 

Pros. 2. The front and side views are 
given to obtain the top view. See that all the 
lines of the object are shown in this view. 

Pros. 3. Note the difference between the 
top views of this figure and the preceding. 

Pros. 4. Remember to represent the in- 
visible lines. 

Pros. 5. Draw the top view without the 
aid of compasses, it being an equilateral triangle. 

Pros. 6. ‘This problem differs from the 
preceding in being a pyramid instead of a prism. 

Pros. 7. Do not use the compasses for 
the construction of the hexagon. 

Pros. 8. Although the front view alone 
is given, it is better to draw the top view first. 





PROJECTION 129 


OBTAIN TOP 
VIEW OF WEDGE 


OBTAIN SIDE 
IEW OF PRIS 


PROB. 4. 
OBTAIN Top piew OF DIAMETER OF 


RECTANGULAR 


AMID 
PYR P 


OBTAIN FRONT AND SIDE VIEW 
OF PENTAGONAL PRISM 
2+ HIG 


OBTAIN FRONT AND SIDE VIEWS 
OF HEXAGONAL PRISM 
24"HIGH 


130 PROBLEMS 


PROBLEMS 9 to 16 are similar to the preceding and may be substituted for them when it is 
desired to omit a graphic statement. Draw three views. The space required is 5’ x T'’, which 
allows four problems to each plate. | 

Pros. 9. Draw a rectangular prism pu long. The bases measure {!’ by 12", and are 
parallel to the profile plane. ‘The prism is resting on one of its narrow faces. 

Pros. 10. Draw a wedge, the front view of which is an isosceles triangle having a base 
of 23” and a height of 23’’.. Length of wedge 14”. 

Pros. 11. Draw a square pyramid resting on its base with two edges of the base making 
angles of 15° with V. The base is 13/" square, and the height of the pyramid is 2,3,"’. 

Pros. 12. Draw a pentagonal prism resting on a lateral face which is parallel to H, and 
the bases perpendicular to V. The bases are inscribed in a circle 17’ in diameter, and the 
sides are 22’ long. 

Benes 13. Draw a triangular prism resting on a face parallel to H, and the bases parallel 
to V. The bases are equilateral triangles having sides of 14/’.. The sides of the prism are 
24!’ long. 

Pros. 14. Draw a triangular pyramid. The base is an equilateral triangle having sides 
of 13" and parallel to H. One edge of the base makes an angle of 15° with V. The pyramid 
is 24” high. | 

Pros. 15. Drawa hexagonal prism resting on a face parallel to H, and the bases parallel 
to V. The faces are $!’ x 21". 

Pros. 16. Draw a hexagonal pyramid 21”’ high. Its base is parallel to H, with two of its 
edges perpendicular to V. The edges of the base are 13”. 


ORTHOGRAPHIC PROJECTION 131 


Pros. 17. Draw a hollow triangular prism, the faces of which are 4” thick. Draw the 
side view first. 

Pros. 18. It is required to represent three views of the preceding object when turned 
around on its base. ‘This will change the angle at which the top view is drawn, but does not 
alter the relation of the lines to each other; therefore, the top view may be copied from Prob. 
17. In drawing the front and side views observe 


that all points of the object retain their former re ad 
. . . _ COPY TOP VIEW FROM 
height, which may be obtained from the front Cuiteue tet: PRECEDING PROBLEM BUT WITH 


THE LONG EDGES AT AN ANGLE 


P . . 39 i 
and side views of Prob. 17. Use care to repre Data eiey t con 


sent the invisible lines of the object. 
Pros. 19. The object to be represented is 
similar to the preceding, save that the ends are 


OBTAIN FRONT AND SIDE VIEWS 


PROB, I9. PROB. 20. 


beveled and the triangular space does not pass BETAMCOr VIEW COPY TOP VIEW FROM 
. ° e : PRECEDING PROBLEM BUT WITH 
entirely through the prism. Omit the drawing Ey Ry OP 
> OF 30 WITH G.L. 
of GL, and the traces of P. Art. 96, page 66. 
The drawing of the projection lines may be SEEN CHORD Akron Vine 





omitted also. Use care to project all of the 
invisible lines of the object. | 

Pros. 20. It is required to represent three views of the preceding object when turned 
around on its base. The construction is similar to that of Prob. 18. In this case the center 
lines will no longer serve for the GL and traces of P. Use care to project all of the invisible 
lines of the object. 


132 


DRAW THREE VIEWS | DRAW THREE VIEWS 
OF THE OBJECT OF THE OBJECT 
WHEN REVOLVED WHEN REVOLVED 


PROBLEMS 
21 TO 32 


FROM THE POSITION | FROM THE POSITION 
OF FIG. 1, 30° TO THE | OF FIG. ee 25° ABOUT 
LEFT ABOUT AN AXIS|AN AXIS PERPENDIC- 
DRAW THREE VIEWS |PERPENDICULAR TO V 


ULAR TOH 


FIG. 3. 


DRAW THREE VIEWS | DRAW THREE VIEWS | DRAW THREE VIEWS 
OF THE OBJECT OF THE OBJECT OF THE OBJECT 


WHEN REVOLVED WHEN REVOLVED WHEN REVOLVED 
FROM THE POSITION |FROM THE POSITION FROM THE POSITION 
OF FIG.|I, 20 FORWARD OF FIG.2, 15 FORWARD] OF FIG. 5, 35° ABOUT 
ABOUT AN AXIS PER- | ABOUT AN AXIS PER- | AN AXIS PERPENDIC- 

PENDICULAR TO P PENDICULAR TO P ULAR TO H 


FIG. 4. FIG. 5. FIG. 6. 


ka Peor. 24% 
altitude of Tie 


a PROB. 22: 
altitude of 13!’. 


TOP VIEW OF FIG. Is 


ob 
RE 


PROB? ?23. 
altitude of 17! if 


TOP VIEW OF. FIG. I. 
1 
1 


F 


Pros. 24. 
altitude of 13". 


TOP VIEW OF FIG. |. 





PROBLEMS 


145. Objects oblique to the coordinate planes. 
A careful study of Art. 97, page 67, must be 
made previous to the solution of Problems 21 
to 32 inclusive. The plate will be divided into 
six rectangles as in the accompanying figure, 
the division lines being drawn in pencil only. 
Six positions of three views each will be re- 
quired in each case. Omit the drawing of pro- 
jection and shade lines. If difficulty is found 
with the problems, number the points as in Fig. 
107, page 65, but use care to retain the same 
number for each point throughout the problem. 


In the required positions draw a rectangular pyramid having an 
In the required positions draw a rectangular prism having an 


In the required positions draw a triangular pyramid having an 
The base is an equilateral triangle. 


In the required positions draw a triangular prism having an 
The bases are equilateral triangles. 


ORTHOGRAPHIC PROJECTION 


Prop. 25. In the required positions draw a pentagonal pyramid having an 
altitude of 13’. Diameter of circumscribing circle of base 13’. 


Pros. 26. In the required positions draw a pentagonal pyramid having an 
altitude of 13’... Diameter of circumscribing circle of base 13'’. 


Pros. 27. In the required positions draw a pentagonal pyramid having 
an altitude of 13’. Diameter of circumscribing circle of base 12”. 


Pros. 28. In the required: positions draw a hexagonal pyramid having an 
altitude of 12’. 


Pros. 29. In the required positions draw a hexagonal pyramid having an 
altitude of 17’. 


Pros. 80. In the required positions draw a wedge having an altitude 
rot 12. 


Pros. 31. In the required positions draw a wedge having an altitude 
of 14”. 


Pros. 32. In the required positions draw the frustum of. a rectangular 
pyramid having an altitude of 12”. 


4 


138 


TOP VIEW OF FIQ. I, 


‘TOP VIEW OF FIQ. |. 


TOP VIEW OF FIG. | 


TOP VIEW OF FIG. I. 
a Fs 
ora 


i 


TOP VIEW OF FIG. |, 
oe 
ae 


A 
TOP VIEW OF FIG, I. 


k- 13 


Das) i 


TOP VIEW OF FIG. I. 
ca be 
yanmar, 
ee] + 
5 aE 7 


TOP VIEW OF FIQ. I. 


134 PROBLEMS 


PROB. 33. 
OBTAIN THE PROJECTION 
ON THE AUXILIARY PLANE 


PROB. 35 
HEIGHT OF, 3” 
PRISM 


8 


REVOLVE CONE 30 ABOUT 
VERTICAL AXIS 
OBTAIN 3 VIEWS 








PROBLEMS 33 to 37 require a knowledge of 
the use of auxiliary planes. Art. 98, page 70. 

Pros. 33. The auxiliary plane being par- 
allel to the plane of the right-hand end of the 
object, the projection on that plane will be a 
true representation of that surface. Other lines 
and surfaces will be foreshortened. 

Pros. 34. In this problem the auxiliary 
plane makes an angle of 30° with H, and the 
projection on this plane is to be made in place © 
of the top view. 

Pros. 35. Observe that the prism has a 
triangular hole extending through it. 

Pros. 36. This is similar to the preceding, 
save that the prism is hexagonal instead of tri- 
angular. A circular hole, 14!’ diameter, extends 
from base to base. 

Pros. 87. Study Arts. 99 and 100, pages 
72 and 73, before solving this problem. Having 
obtained the base of the cone in the manner 
directed, locate the vertex of the cone. The 
tangents to the base drawn from the vertex will 
be the contour lines of the cone. 


ORTHOGRAPHIC PROJECTION Veh 


146. Special problems in projection. ‘These problems require a space of 5!’ x 7!’. Three 
views are required in each case, and all invisible as well as visible lines should be shown on 
each view. Leave all construction lines in pencil. All polygons are regular polygons. 

Pros. 388. Draw the frustum of an octagonal pyramid having its base parallel to H and 
two of its edges making an angle of 30° with V. The diameter of the circumscribing circle of 
its lower base is 14’, and of the upper base, 14". The altitude is 17’. 

Pros. 39. Revolve the pyramid of Prob. 88, 30° to right about an axis perpendicular 
to V. 

Pros. 40. Draw a pentagonal prism resting on one of its faces and having its lateral 
edges at an angle of 223° with V. Diameter of circumscribing circle of base 1}/’.. Length of 
prism 24’. 

Pros. 41. Draw an equilateral triangular prism resting on one of its faces, and its lateral 
edges making an angle of 15° with V. The edges of the base are 13’’, and the length of the 
prism is 24’. There is a triangular hole extending through the bases and making the thickness 
of the sides 4’. 

Pros. 42. Draw a cylinder with its axis parallel to V and at an angle of 60° with H. 
The diameter of the base is 13’’, and the length of cylinder, 2}/’.. Obtain the ellipses by the 
method of trammels. 

Pros. 43. Draw an equilateral triangular pyramid having an altitude of 24’, and the 
edges of the base 17/’.. The base makes an angle of 30° with H and one of its edges is per- 
pendicular to V. 

Pros. 44. Revolve the pyramid of Prob. 43, 45° forward about an axis perpendicular 
toRk: 


136 PROBLEMS 


Pros. 45. Draw a box having the following outside dimensions. Length 2!, width 13”, 
depth, including cover, 1/’.. Thickness of material 4’... The long edges of the box are parallel 
to H and make an angle of 30° with V. The cover is hinged on long edge and opened 30°. 

Pros. 46. Draw a pyramid formed of four equilateral triangles having 22” sides. The 
base is parallel to H and one of its edges makes an angle of 30° with V. 

Pros. 47.. Draw a rectangular surface, 14!’ x 28’, in the following positions : The short 
edges parallel to H and making an angle of 75° with V; the long edges making angles of 15°, 
80° and 45° with H. Art. 102, page 76. 

Pros. 48. Revolve the surface from the positions required in Prob. 47, 15° forward. 

Pros. 49. Draw an isosceles triangle in three positions as follows : The base lying on V 
and inclined at an angle of 30° with H. ‘The altitude making angles of 90°, 30° and 15° with V. 
The base of the triangle is 14’’, and the altitude 23. Art. 102, page 76. 

Pros. 50. Draw the same triangle revolved from the positions in Prob. 49, 30° in either 
direction about a vertical axis. | 

Pros. 51. Draw an isosceles triangle in the following positions: The base parallel to H 
and making an angle of 60° with V. The altitude making angles of 45° and 60° with H. The 
base of triangle is 2’, and the altitude 21’’.. Art. 102, page 76. 

Pros. 52. Revolve the same triangle from the positions in Prob. 51, 30° backward. 

Pros. 53. Draw an octagonal surface inclined at an angle of 60° with H, two of its edges 
being parallel to H and making angles of 15° with V. ‘The diameter of circumscribing circle 
is: 24"! Arte 102; page 76. 

Pros. 54. Draw a hexagonal surface inclined at an angle of 45° with V, two of its 
edges being parallel to V and making angles of 30° with H. The long diameter of the hexagon 
is 24". Art. 102, page 76. 


ISOMETRIC PROJECTION s¥6 


Pros. 55. Draw the projections of a line located as follows: The left-hand extremity 
of the line is 3 behind V and 7’ below H. The right-hand extremity is 14/’ behind. V, and 
13" below H. The H projection of the line makes an angle of 30° with V. Find its length by 
revolving it parallel to H, Vand P. Art. 101, page 74. 

Pros. 56. Draw the projections of a lne of which the left-hand extremity is 1'’ behind 
V and 12” below H. The right-hand extremity is 3’’ behind V and 3"’ below H. The H pro- 
jection makes an angle of 15° with V. Find the length of the line by revolving it into the 
planes of projection by the second method. Art. 101, page 75. 


147. Isometric projection problems. Study Chapter VI, page 78. The problems require 
a space of 5’ x 7'’.. Omit the invisible lines in inking. : 

Pros. 1. Make the isometric drawing of a 2’ cube. Art. 110, page 80. Inscribe circles 
on the upper and right-hand faces, the former by the exact method and the latter by the 
approximate method. Art. 112, page 81. From the left-hand lower corner of the left-hand. 
face draw lines making angles of 30°, 45° and 75° with the lower edge. Art. 113, page 82. 

Pros. 2. Make the isometric drawing of the frustum of a pyramid, the lower base being 2”, 
and the upper base 13/’ square. Height 14’’. Inscribe a circle on the upper base using the ap- 
proximate method. Locate the front lower corner in the center and 1'’ from lower margin. 

Pros. 3. Make the isometric drawing of a pentagonal plinth surmounted by a cylinder. 
The sides of the pentagon are 2" and the height of the plinth is 3’’.. Art. 71, page 47. The 
cylinder is 2!’ in diameter and 1” high. Art. 111, page 80. 

Pros. 4. Make the isometric drawing of a box with cover opened through an angle of 
120°. The outside dimensions are: length 21/’, width 13/’, depth #/’.. Thickness of material 
is 4’. Locate the front lower corner in the middle of the space and 4/’ from lower margin. 


138 





PROBLEMS 


or icpan 


ska 
ea! 


PROB. 


6 


Pros. 5. Make the isometric 
drawing of the bearing illustrated, 
locating the upper corner at point A. 

Pros. 6.. Make the isometric 
drawing of a hexagonal bolt. ‘The 
center line may be parallel with either 
of the isometric axes. Art. 115, p. 83. 

Pros. 7. Make the isometric 
drawing of the pieces illustrated, and 
a second isometric drawing of the 
upright block showing the cuts neces- 
sary for making the required fits. 
The lowest portion of the upright 
block will be located at A in the first 
ease and B in the second. 


Pros. 8. Make the isometric drawing of the connecting-rod strap, the scale to be 3/’=1 ft. 
In drawing the curves observe the directions of Art. 115, page 83. 

Pros. 9. Make the isometric drawing of the framing details illustrated. The dimensions 
of the materials are given below. The space required is 10’ x 14’’. Draw toa scale of 13/’=1 ft. 


Sills, 613". 
Post, AS Bll 
Brace, AM Di, 


Window studs, 4x4". 


Studs, 2" x4", 12" on centers. 
Floor joists, 2x8", 12" on centers. 


Under flooring, 7" x10". 
Upper flooring, 7x 6". 


Grounds, #3!'x 2". 

Laths, 3" 11" x 48", 3" space. 
Plaster, 3" thick. 

Baseboard, 7x10", including cap. 


‘ ISOMETRIC PROJECTION 139 


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BASE-BOARD 
















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a2 


v/, 
Z 
FLOOR JOIST 2X6. - 


i 








LLLLLLL LION a Roope FLOOR ho epee te 
Cpoeneeneee _ UNDER FLOOR 





| 
: FLOOR JOIST 


FLOOR JOIST 2x8 


i 


SILL. 





WINDOW STUD 


LAYOUT OF SHEET. 


Aen 






ET FT a 
LABBawaa’ 


DETAILS FOR ISOMETRIC DRAWING 


V|_ FLOOR voisT. OF FRAMING PLANS 


SSX y 
EY) 


140 


WE/EH7S 
PULLEY 


NS TN 












QQ e{} 2 
a AN Sy 
Nt 





PROBLEMS 


Pros. 10. 
detailed sketches. 


Make the isometric drawing of a window from the 
The dimensions of the materials are given below. 


Show all that is given in the sections. Locate the drawing as in the 


lay-out sketch, observing that the short edge of the plate is hori- 


zontal. 





LAYOUT OF SHEET 


Art. 115, page 83. 


Draw to scale of 3! =1 ft. 


DIMENSIONS OF MATERIALS 


Studs, 

Apron, 

Ground easing, 
Inside architrave, 
Outside architrave, 
Outside casing, 
Outside boarding, 
Pocket, 

Pulley style, 
Parting bead, 
Stop bead, 

Laths, 


Plaster, 


Al x 4", 

"x33", 

3" x 4a", 

gil x 5B 

13"x 3h" 

se ga 

4. 

Qi 4 

i" thick. 

BIN yc BI, 

a" x14". 

3" x 1i"x 48", 
spaced 8", 

3" thick: 


Clapboards, 4" at thick edge, x 54!" x 48", 
laid 34"' to the weather. 


DEVELOPMENT OF SURFACES 141 


148. Development problems. 
ter VII, page 88. 
plete development will be required in each case. 
In general, begin to develop the surface on the 
shortest edge. It will assist the student to a 
better understanding of this subject if the de- 
veloped surface be copied on stiff paper and 
afterwards cut and folded. Observe the order 
prescribed for performing these problems. Art. 
119, page 90. 

Pros. 1. This isa prism having a pentag- 
onal base. ‘The vertical edges being parallel 
to V are seen in their true length on that plane. 
The length of the edges of the lower base can 
be obtained from the top view, and the true shape 
of the upper base will be found by projecting it 
on to an auxiliary plane, as in Art. 98, page 70. 

Pros. 2. Solve as for Prob. 1. 

Pros. 8. In this and the following prob- 
lems do not ink that portion of the surface lying 
above the cutting plane. Art. 119, page 90. 

Pros. 4. Observe that the slant edges of 
this pyramid are not of equal length. 


Study Chap- 
Three views and the com- 


BOB Me oe viEWion PROB. 2 
| PENTAGON 


- HEXAGONAL PRISM 
— INSCRIBED Nat | REVOLVED 
| 2 CIRCL SECTION 


easy. aon 


SIDE VIEW 
| OBTAIN 
igharanannns 


DEVELOPMENT DEVELOPMENT 


PROB. 3 PROB. 4 
| 


=| REVOLVED 
SECTION | SECTION 


REVOLVED 


OEVELOPMENT DEVELOPMENT 





142 PROBLEMS 


PROB. 5 ’ PROB. 6 
TOF VIEW OF TOP VIEW OF 


s QUARE PYRAMID SQUARE tf atl 


REVOLVED REVOLVED 
SECTION ! errr 


SIDE VIEW 


ex 
| 
| 


DEVELOPMENT 


DEVELOPMENT DEVELOPMENT 





Pros. 5. This is a square pyramid, the 
diagonal of the base being 2’’.. Observe that the 
base is cut by the plane, and one of the inclined 
edges will not appear on the completed develop- 
ment. In developing the surface open it on 
this line. 

Pros. 6. This differs from the preceding 
in that the base is not cut by the plane, and the 
position of the pyramid with respect to V is 
changed. 

Pros. 7. The surface cut by the plane 
will not be symmetrical with respect to the 
center line as in the preceding problems. Care 
must be used to take no dimensions from the 
top view that may not be represented by lines 
parallel to H, the vertical projection of which 
will be parallel to GL. 

Pros. 8. This is similar to Prob. T, only 
the position of the pyramid with respect to V 
being changed. 


DEVELOPMENT OF SURFACES 


Pros. 9. The cutting plane makes an 
angle of 40° with H, and the auxiliary plane 
must be at the same angle, the projecting lines 
being drawn perpendicular to it. 

Pros. 10. This is similar to Prob. 5, the 
base being cut by the cutting plane. In devel- 
oping the surface open it on the uncut edge. 

Pros. 11. Theslant edges of this pyramid 
being of unequal length must be obtained sepa- 
rately. One of these edges, AD, is shown in 
its revolved position at AD’. One-half of the 
development is also shown, and the method and 
order for drawing the lines indicated by the 
numbers. Thus, the line AB is drawn first, 
and then arcs 3 and 2 described from its ex- 
tremities, A and B, with radii equal to the true 
lengths of AC and BC: this determines the 
point C. In like manner the remaining points 
and lines are found. « 

Pros. 12. The development of a cylin- 
der is required. Arts. 120 and 121, page 92. 
Employ twenty-four elements in obtaining the 
curve. 


° 148 


PROB. 9 te PROB. 10 


TOP VIEW OF , 
HEXAGONAL PYRAMID 


| PENTAGON 
INSCRIBED IN 
i 2-\"CIRCLE 
as 
 REVOLVED 
| SECTION 


+ 
H REVOLVED 
SECTION 


ee 


oy 


nN 
| SIDE VIEW SIDE VIEW 


| 





d lo 
<a> 


DEVELOPMENT DEVELOPMENT 


PROB. I2 


REVOLVED 
SECTION 


DEVELOPMENT 





144 





PROBLEMS 


PROB. |3 PROB. 14 


REVOLVED aa REVOLVED 
SECTION . . SECTION 





SIDE VIEW 





DEVELOPMENT DEVELOPMENT 


PROB. I5 


Pros. 4. Solve without auxiliary planes. 


Pros. 138. Use twenty-four elements for 
obtaining the development. Art. 122, page 94. 

Pros. 14. This being an elliptical cone, 
the elements will be of unequal length. 

Pros. 15. The cone is cut by four planes, 
CG, CB, CF, CE. Determine the top view of 
each and their projection on planes to which 
they are parallel. ‘Test the latter as follows: 
Determine the axes and foci of the ellipse, and 
test eight points by the first method, Art. 80, 
page 52. Test by trammels also. Test the 
parabola by the second method, Art. 87, page 
58. The cutting plane, CI, is parallel to an 
element of the cone. Test the hyperbola by 
the second method, Art. 91, page 59. The ver- 
tex of the cone bisects the transverse axis. 


149. Intersection problems. Chap. VIII. 
Pros. 1. Assume twenty-four equidis- 
tant elements on the small cylinder. 
Pros. 2. The cylinders are tangent. 
Use care in obtaining the limiting points. 
Pros. 8. Use auxiliary planes. Page 98. 
Art. 125, p. 100. Develop oblique cylinder. 


INTERSECTION 


DEVELOPMENT OF CYLINDER A 


SIDE VIEW 


DEVELOPMENT OF CYLINDER B 


PROB. 2 


DEVELOPMENT OF CYLINDER A 


| SIDE VIEW 


’ 


DEVELOPMENT OF CYLINDER B 





OF SURFACES 


TOP VIEW 


DEVELOPMENT 
OF CYLINDER A 


DEVELOPMENT 
OF CYLINDER 





145 


146 PROBLEMS 


DEVELOPMENT OF DEVELOPMENT OF 
EQUILATERAL TRIANGULAR EQUILATERAL TRIANGULAR 
PRISM 


TOP VIEW 


DEVELOPMENT OF 
HEXAGONAL PRISM HEXAGONAL PRISM A 





Pros. 5. It is required to find the inter- 
section of a cylinder and prism without using a 
side view. Use cutting planes parallel with V. 
The base of the prism will have to be revolved 
in order to complete the top view and enable the 
intersection of the cutting planes and prisms to 
be determined. 

Pros. 6. To determine the intersection 
of a hexagonal and a triangular prism. This is 
similar to Prob. 5, save that a prism is substi- 
tuted for the cylinder. In all cases of inter- 
section between prism and prism, it is only 
necessary to find the point of intersection of 
each edge of both prisms with a face of the 
other prism. Art. 126, page 101. 

Pros. 7. The lines of intersection on the 
top view are to be completed, and the front 
view with its lines of intersection are required. 
Develop the prism only. 

Pros. 8. Determine the intersection of 
the oblique and vertical hexagonal prisms and 
develop the latter. Art. 126, page 101. 


SPIRALS AND HELICES 147 


150. Spirals, screw-threads and bolt-head problems. Study Chapter IX. Page 102. 
Problems 1 to 9 inclusive require a space of 43’. A graphic statement is made for the prob- 
lems on screw-threads and bolt-heads. 

Pros. 1. Draw an equable spiral with a pitch of 11/. Describe 13 revolutions of the 
radius vector. Art. 128, page 102. Sketch the 


curve, and ink as directed in Art. 15, page 12. Lea aad 
Pros. 2. Draw the involute of a 3” square. --3 }-—> 
j i! : SECTION OF 
Art. 130, page 104. 2 SQUARE THREAD 14 3 Sa es 
Pros. 3. Draw the involute of an equi- aoa ute at [THREAD | 


lateral triangle having 3!’ sides. Art. 180, 
page 104. 

Pros. 4. Draw the involute of a right Branckaghiy enon 
line 4’ long. Art. 130, page 104. pally ; ANGLE OF, V 90" 

Pros. 5. Draw the involute of a hexagon 
having sides of #/’. Art. 130, page 104. 

Pros. 6. Draw the involute of a circle 
having a diameter of 1’... Art. 130, page 104. 

Pros. 7. Draw a right-hand helix of 14/’ pitch, 2’’ diameter and 23’ length. Art. 131, 
page 104. | 

Pros. 8. - Draw a left-hand double helix of 13/’ pitch, diameter 13’, length 21". 

Pros. 9. Draw a right-hand conical helix. Pitch 1}!’, diameter of cone 2’, height 17/’. 

Pros. 10. The diameter and pitch being the same in both cases, but two templets are re- 
quired, one for the outer and one for the inner helix. Art. 132, page 106, and Art. 186, page 110. 





148 PROBLEMS 


R.H. THREAD 
24 aye 








5 ° 
3 PITCH 60 V 








{—_ - 
. R.H. DOUBLE 
. | . . CA . sQ! it P. 


Noo 
— gt =e a 

2 : itp 
R.H.SINGLE y 


, 
haa S iciecae Same 





; Z 
Ce oe oe 





Pros. 11. The first four examples are of 
conventional V threads, Art. 183, page 108. 
The second four are conventional square 
threads. Art. 186, page 110. The last four 
are to be drawn by the methods illustrated by 
Figs. 177 to 180, page 109. Make the pitches 
about the same as those in the illustrations, 
estimating the spaces by the eye. Distinguish 
clearly between the single and double threads. 

Pros. 12. Study Arts. 187, 188 and 139 
before attempting this problem. The propor- 
tion and character of the heads and nuts should 
be so well understood that reference to the text 
will be unnecessary. ‘The diameters are given, 
and the sketch shows the character of the bolt, 
whether rounded or chamfered. Observe every 
detail and see that the dimensions are standard. 
Draw the rounded heads and nuts before the 
chamfered type. 

For the further consideration of this subject, 
the student is referred to the chapter on Bolts 
and Screws, in ‘Machine Drawing” of this 
series. 


MISCELLANEOUS PROBLEMS 


151. Miscellaneous problems. These 
examples of a practical character are de- 
signed for pencilling and inking practices 
which may be used early in the course. 
Problems 1 and 2 may be substituted for 
some of the “examples for practice,” page 
117 and following. They will afford an 
excellent practice in lettering and figuring. 
Problems 3 and 4 hkewise require very 
little knowledge of projection and serve 
as an excellent practice in penmanship. 
Full instruction is given with each sketch 
and the proper scale is specified for a 
10" x 14" plate. The shading of the 
drawing is left to the discretion of the 
instructor. ‘The drawings should be cor- 
rectly figured, and the title in proper form. 

The author does not recommend the 
making of machine drawings, save as above, 
until the student has acquired a thorough 
working knowledge of projection, and has 
been taught something of the notation and 
idiomatic use of applied graphics. 





149 


PROB.| 








ECCENTRIC. eae 
DRAW TWO FULLVIEWS. 
SHADE AND DIMENSION LINES REQUIRED, 


150 PROBLEMS 


TWO COMPLETE SECTIONS OF THE FIRST 
AND SECOND FORMS OR THE \FIRST ANDO 
THIRD FORMS MAY BE Ba alsen INA 
fA C 10/4 SPACE. | 









STANDARD, SECTION OF 


Z£OC al D SEWER 








Ca ae YQ: 
Ww 
480 
. ~ : 
AN SI 
RII; | 
3 
aX ID 
So: 
18 | 
DRAW COMPLETE SECTION DRAW COMPLETE SECTION DRAW COMPLETE, SECTION 
SCALE /°=/F7. SCALE 22/FT. SCALE 2 =/FT. 


a 





MISCELLANEOUS PROBLEMS 15 


ONE WROUGHT JPON. 


BoRDER £/NE 





HAN) LEVER FOR CONTROLLING CYL/NDER 
VALVES U.S. COAST DEFENSE VESSEL MONTEREY 


152 PROBLEMS 








2 


Wa 


ra 


SO BARS % 





| 
| 
| 
| 
| 
| 
| 


| 
| 
| 
| 
| 
| 
eds | 
| 
| | 
| | 


Pros. 4. A sketch is given for the Commutator of a 10 K.W.D.C. Dynamo, and 
it is required to make a full-size sectional drawing of the complete commutator. 








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